1 Meeting 1 Introduction to Functions Part 1 Graphing Points on a Plane (REVIEW) A plane is a flat, two-dimensional surface. We describe particular locations, or points, on a plane relative to two number lines (commonly called the coordinate axes), a horizontal axis and a vertical axis. The picture below shows the horizontal and vertical axes: Notice that the two axes intersect at the number 0 on both number lines. Most of the time, this will be the case. Every point on this plane can be identified/located/described by a pair of numbers called coordinates. The first coordinate is determined from the horizontal axis and indicates how far left or right of the vertical axis a point is located. The second coordinate is determined from the vertical axis and indicates how far above or below the horizontal axis a point is located. For example, the pair (2,-3) represents the point that is 2 units to the right of the vertical axis and 3 units below the horizontal axis. The picture above shows this point. The point given by (0, 0) lies right at the intersection of the coordinate axes and is commonly referred to as the origin. For each of the points shown below, give the coordinate pair describing that point. Part 2 What is a function? The figure to the right depicts a function: Input Input Function A function is a process whereby each numerical input is assigned at most one numerical output. Output
2 2 Example 1: Consider the following process: Take any given number and square it. Then subtract 5. This process can be summarized/represented/displayed/described in four different ways: 1. Numerical Representation (table) Inputs Outputs Does this process describe a function? Yes. Each input yields exactly one output. Thus, it is a function. 2. Pictorial Representation (flow diagram) Input 1. Square the input. 2. Subtract 5. Output 3. Conceptual/Algebraic Representation (equation) When working with functions, there will always be 2 varying quantities, or variables. One variable is dependent on the other. Independent Variable = Input Dependent Variable = Output Rather than always writing Input and Output, we usually give the variables letter names. For example, the equations all represent exactly the same function. The letters B, t and x all represent the input (i.e. independent variable) of this function, while the letters S, u and y all represent the output (i.e. dependent variable). You sometimes hear the phrase y is a function of x. This just means that the variables x and y represent the input and output respectively of a certain function. In our example, would we say B is a function of S or S is a function of B? S is a function of B
3 3 4. Graphical Representation The graph of a function displays the input/output pairs for a function. Each input/output pair is represented by a point in the plane having coordinate pair. Consequently, the horizontal axis generally corresponds to the inputs of a function, while the vertical axis corresponds to the outputs. Consider the function given by the equation table.. Complete the following Input (x) Output (y) Coordinate Pair (-4, 12) 3 5 (-3, 5) -2 0 (-2, 0) -1-3 (-1, -3) 0-4 (0, -4) 1-3 (1, -3) 2 0 (2, 0) 3 5 (3, 5) 4 12 (4, 12) Notice that the axes shown below are labeled x and y. If we represent our function by the equation, then the input and output variables are represented by the letters x and y respectively, and the axes are referred to as the x-axis and y-axis. Plot the points given by the above coordinate pairs on the axes provided. The picture above is not the full graph of the function. The plotted points correspond only to the integer inputs between -4 and 4.To draw the full graph, we would need to continue plotting points for ALL possible input values, including non-integers. However, from the points you have plotted above, you can see a pattern or trend that should enable you to predict what the full graph looks like. The picture below shows the completed portion of the graph lying between -4 and 4.
4 4 The intercepts of a graph are the places where a graph passes through one of the axes. The graph above passes through the x-axis at two different points and passes through the y-axis once. Give the approximate coordinates of the intercepts. Points on x-axis: (-2, 0) and (2, 0) Point on y-axis: (0, -4) The two values of x, 2 and -2, are called the x-intercepts or horizontal intercepts of the graph. The value y = -4 is called the y-intercept or vertical intercept of the graph. Example 2: Draw a graph of a function that has a horizontal intercept at -3 and a vertical intercept at 4. Example 3: Determine the horizontal and vertical intercepts for the function given by the equation. Hint: Recall that the output must equal 0 when the graph intersects the horizontal axis, and the input must equal 0 when the graph intersects the vertical axis. The horizontal intercept will be the value of t for which y = 0. In other words, we need to determine the value of t for which 4t - 16 = 0. This suggests that 4t must equal 16, and consequently, t = 4. The vertical intercept will be the value of y for which t = 0. If we substitute t = 0 into the equation y = 4t - 16, we see that y = 0-16 = -16. So, the horizontal intercept of this function is t = 4 and the vertical intercept is y = -16. Example 4: Answer each of the following questions: (a) Can the graph of a function have multiple horizontal intercepts (i.e. multiple input values for which the value of the output variable is 0)? Yes. The graph on the previous page has two horizontal intercepts. (b) Can the graph of a function have multiple vertical intercepts (i.e. multiple output values when the value of the input variable is 0)? No. A function can have at most one output for each input value. So, when the input of the function equals 0, the output variable can have at most one value.
5 Part 3 Interpreting Algebraic Representations of Functions 5 Example 5: Consider the function whose algebraic representation is. (a) Independent variable (input) = v Dependent variable (output) = z (b) Is z a function of v or is v a function of z? z is a function of v. (c) Make a flow diagram for this function. v z (d) Complete the following input/output table for this function: v z When you ve finished parts (a-d), compare your results with others sitting around you. Do you arrive at the same results? Many of you likely had different flow diagrams and different outputs for this function. Example 5 shows how easy it is to misunderstand a function s algebraic representation! Misunderstandings commonly occur when we need to use several mathematical operations (addition, multiplication, subtraction, division, exponentiation, etc.) to obtain the output of a function. When we work with functions, we must completely understand the order in which these operations are to be done, or we ll get the WRONG outputs. The box below summarizes the order in which we agree to perform operations: When working with an expression involving multiple operations, the operations should be performed in the following order: 1. Operations within grouping symbols MUST be completed first. Grouping symbols include 2. Perform any required exponentiation (i.e raising numbers to powers). 3. Perform any required multiplication or division from left to right. 4. Perform any required addition or subtraction from left to right. Before returning to example 3, let s practice these operations.
6 6 Example 6: Simplify each of the following expressions. (a) =15 21= 6 (b) (c) ( ) 1 4( 2 2 6) =1 4( 4 6) =1 4( 2) =1 ( 8) =1+ 8 = 9 (d) ( 4 2) ( 4 2) 2 = 5 9 ( 8) 2 = = 19 (e) (f) This is tricky. Perhaps a helpful way of thinking about this example is to consider the fact that is the same as. How would you simplify? You would perform the exponentiation first, and then perform the subtraction. In other words,. (g) (h) Now that you ve had some practice applying the order of operations ideas, let s re-examine example 5. Example 5 (continued): So, what is the correct way to interpret? The correct flow diagram is: v 1. Square v. 2. Multiply by 4. z The outputs are obtained as follows: v 1 z
7 7 3-2 Example 7: Consider the function defined by the equation. (a) Independent variable (input) = u Dependent variable (output) = r (b) Make a flow diagram of this function. u 1. Square u. 2. Multiply by Subtract u. 4. Add 1. r (c) Complete the following input/output table for the function : u x t r Part 4 Function Names and Notation Input and output variables of functions are generally represented by letters. In addition, we often assign letter (or word) names to the functions themselves. One benefit of doing this is that a particular function can be referenced more easily on occasions when we happen to be working with multiple functions at the same time. Consider, for example, three functions given by the equations p = 4( 1.6) t, r = 5 3t, and s = t 2 1 We could call the first function f, the second function g and the third function h. Then, anytime we need to refer to one of these functions, we simply write f, g, or h instead of writing the entire equation. A second benefit of giving a function a name is that input/output pairs can be quickly and easily represented. For example, consider the function g above. (In other words, we re talking about the function r = 5 3t). Notice that when t =1, r = 5 3(1) = 2. Instead of writing the lengthy English sentence When t =1, it follows that r = 2.
8 8 mathematicians will simply write g(1) = 2 instead. Similarly, for the function h above (i.e. s = t 2 1), instead of writing When t = 0, it follows that s = 1, we simply write h(0) = 1 instead. In general, for a given function, we can write every input/output pair as follows: Function name( Input ) = Output The function equations themselves can also be written using this notation as follows: So, in other words, p = f (t), r = g(t) and s = h(t). Example 8: Consider the functions given by and. Determine each of the following. (a) (b) f (t) = ( ) t, g(t) = 5 3t and h(t) = t 2 1. (c) (d) (e) (f) (g) (h) because because Example 9: Consider the function given by. (a) Which letter represents the input of this function and which letter represents the name of the function? The letter x represents the input variable and the letter f represents the name of this function.
9 9 (c) Determine. (d) Determine the value of x for which. In this problem, you are asked to determine the value of the input variable x for which the output value equals -2. In other words, find the value of x for which. With some experimenting, you see that x = 2 works, since (d) Determine the vertical intercept for this function. The vertical intercept is the value of the output when the input x = 0. Since, the vertical intercept is y = -6. (e) Determine the horizontal intercept for this function. The x-intercept equals the value of x for which the output value of the function equals 0. In other words, we have to find the value of x for which. Setting and doing some experimenting, you find that x = 3 works, since. Example 10: Shown below is a portion of the graph of a function y = p(x). What is the name of this function? What letter represents the input? What letter represents the output? Function name = p Input = x Output = y Use the graph for parts (a) (g) below. (a) Determine p( 2). p( 2) refers to the output of the function f when the input variable x equals 2. From the graph, we see that p( 2) 16. That is, p( 2) is approximately 16. (b) Determine p(2). p(2) refers to the output of the function f when the input variable x equals 2. From the graph, we see that p(2) = 0. (c) If, what must x equal? Here, we are told that the output y equals 10 and we are asked to find the value of the corresponding input x. From the graph, we that. (d) If p(x) = 40, what must x equal?
10 10 Here, we are told that the output equals 40 and we are asked to find the value of the corresponding input x. From the graph, we that. (e) If p(x) = 0, what must x equal? Here, we are told that the output equals 0 and we are asked to find the value of the corresponding input x. From the graph, we that there are three possible values for x, namely. (f) Determine the x-intercepts and the y-intercept. The x-intercepts are the values of the input x for which the value of the output is 0. In part (e), we determined that these values were. From the graph, you can see that the graph passing through the x-axis at these values of x. The y-intercept is the value of the output y when the input variable x equals 0. In other words, the y-intercept equals p(0). From the graph, we see that p(0) = 0. You also see that the graph passes though the y-axis when y = 0. So, the y-intercept is y = 0.
11 Section1 Homework Assignment One table below shows input/output pairs for a function and the other table shows input/output pairs for a process that is NOT a function. Which table describes a function? Why does the other table NOT describe a function? Hint: Consider how many outputs each input has. Input Output Input Output One curve below is the graph of a function and the other curve can NOT be the graph of a function. Which curve is the graph of a function? Why is the other curve NOT the graph of a function? 3. Consider the following process: Multiply a number by 3 and then add 2. (a) Draw a flow diagram for this function. (b) Complete the following input/output table: Input Output (c) Determine an algebraic representation (i.e. an equation) for this function. (d) Use your input/output table to Sketch 4 points on the graph of this function. What do you think the complete graph looks like? 4. Consider the following process: Subtract 4 from a number and then multiply by 2. (a) Draw a flow diagram for this function. (b) Complete the following input/output table: Input Output (c) Determine an algebraic representation for this function.
12 (d) Use your input/output table to sketch 4 points on the graph of this function. Does the graph appear to be a line? 5. Each of the following calculations is flawed in some way. Find the mistakes. Then, determine the correct values for each. (a) (b) (c) (d) (e) 6. Simplify each of the following expressions. (a) (b) (c) (d) (e) 7. Each of the equations below is the algebraic representation of a function. (a) The variables u, s, m, z, y and x represent inputs and outputs for these functions. Which variables represent inputs and which represent outputs? (b) Complete the following input/output tables. Be careful to perform the mathematical operations in the correct order! 12 s 6 3 u z 4 2 m x 3 5 y 8. The following two functions are quite similar. Both functions require multiplication by 3 and addition by 4. For each function, draw a flow diagram. What is the major difference between the two functions? Determine the outputs of both functions when. Note that the outputs are different! 9. The following two functions are quite similar. Both functions require multiplication by 5 and squaring. For each function, draw a flow diagram. What is the major difference between the two functions? Determine the outputs of both functions when. Note that the outputs are different! 10. For each of the following, determine the function name, the input and the output.
13 13 (a) (b) (c) (d) (e) (f) (g) (h) 11. For each of the functions that follow, express the given input/output pair in the form Function Name (Input) = Output. (a) Function name = y; Input = 3; Output = 6 (b) Function name = r; Input = 0; Output = -5 (c) Function name = s; Input = -4; Output = 0 (d) Function name = w; Input = x; Output = y (e) Function name = u; Input = t; Output = 5t-1 (f) Function name = p; Input = u; Output = 3(u-5) 12. Consider the function. Notice that when the input of this function is the value 1, the output value is -2. One of the following is the correct way to write this. Determine which one is correct. Why are the other ways incorrect? (a) (b) When x = 1, f = -2. (c) When x = -2, f = 1. (d) (e) (f) 13. If you are being asked to determine u(8), which of the following is true: (a) The input of the function u is 8, and you have to find the output. (b) The output of the function u is 8, and you have to find the input. 14. If you are being asked to determine the value of a for which y(a) = 0, which of the following is true: (a) The input of the function y is 0, and you have to find the output. (b) The output of the function y is 0, and you have to find the input. 15. Consider the functions f, r, and v defined as follows: (a) Determine,, and. (b) Determine and. (c) Determine. (d) Determine. You do not need to simplify your answer. (e) Determine. You do not need to simplify your answer.
14 14 (f) Determine. (g) Determine. (h) Determine. 16. Consider the function. (a) Determine and. (b) By experimenting, determine the value of x for which. (c) Determine the value of x for which. (d) Determine the vertical intercept for this function. (e) Determine the x-intercept for this function. 17. Shown below are portions of the graphs of functions and. Use the graphs for (a) (h) below. (a) Determine,, and. (b) Determine,, and. (c) Determine the value(s) of x for which. (d) Determine the value(s) of x for which. (e) Determine the value(s) of x for which. (f) Determine the value(s) of x for which. (g) Determine. (h) Determine. (i) Determine the horizontal and vertical intercepts for the function f. (j) Determine the horizontal and vertical intercepts for the function g. 18. Which of the following points is on the graph of the function? (a) (0, 0) (b) (3, 15) (c) (-2, 20) (d) (-1, 1) (Hint: Don t attempt to answer this question by sketching a graph of this function! Remember that a point will lie on the graph of a function only when the coordinates of the point represent an input/output pair for the function. So, you need to determine which of the pairs above is actually an input/output pair for the function. ) 19. Input/output tables for three different functions are shown below. Determine algebraic representations (i.e. equations) for each function. First, decide what letters you want to
15 15 use to represent the independent variable and dependent variable. Then, study the input/output tables to see how the outputs relate to the inputs. (a) (b) (c) Input Output Input Output Input Output
16 16 Section 1 Answers to Selected Homework Exercises 1. The table on the right describes a function while the table on the left does not. Note that the table on the left shows the input value 1 having two different outputs, namely 2 and (a) intput 1. Multiply by Add 2. output (b) Input Output (c) If you let x represent the input and y represent the output, then the algebraic representation is. (d) The 4 points are shown below. If you connect the points, you obtain a line. The graph of this function is actually a line. 5. (a) Addition was performed before multiplication. Correction: (b) Because, the mistake was that multiplication was performed before exponentiation. Correction: (c) Exponentiation was performed before the operation in the grouping symbol. Correction: (d) There are two mistakes involving misuse of the symbol. In both cases, the symbol indicates subtraction. Correction: (e) Multiplication was performed before exponentiation. Correction:
17 17 7. (a) The variables s, z, and x refer to inputs, while u, m, and y refer to outputs. (b) s 6 3 u z 4 2 m x 3 5 y 9. For the function, the flow diagram is x 1. Square a number. 2. Multiply by 5. w For the function, the flow diagram is x 1. Multiply by Square the result. u The major difference between these two functions is the order in which squaring and multiplication are done. When, note that and. 10. (a) The function s name is b. The input value is 0, while the output value is (a) (d) 13. Part (a) is correct. 14. Part (b) is correct. 15. (a),, and (c)
18 (a) (e) (g) Since, it follows that. (a),, and (c) Here, you are trying to locate the input that corresponds to an output of 10. From looking at the graph of f, it appears that. (d) Here, you are trying to locate the input that corresponds to an output of 0. From looking at the graph of f, it appears that. (f) Here, you are trying to locate the input that corresponds to an output of 0. From looking at the graph of g, it appears that there are two such input values, namely and. (g) Since, it follows that. (j) The horizontal intercepts are and. The vertical intercept is y = 10.