Section. A Catalog of Essential Functions Kiryl Tsishchanka A Catalog of Essential Functions In this course we consider 6 groups of important functions:. Linear Functions. Polynomials 3. Power functions. Rational functions 5. Trigonometric functions 6. Eponential/Logarithmic functions EXAMPLES:. Linear Functions f() = m+b where m is the slope and b is the y-intercept. Its graph is a straight line:. Polynomials: P() = a n n +a n n +...+a +a +a where a n,a n,...,a,a,a are constants called the coefficients of P() and n is the degree of P() (if a n ). (a) If a n > and n is even, then its graph is For eample, here is a graph of P() = + 3 3 3 + with a = > and even degree=.
Section. A Catalog of Essential Functions Kiryl Tsishchanka (b) If a n < and n is even, then its graph is For eample, here is a graph of P() = 3 + 9 + with a = < and even degree=. (c) If a n > and n is odd, then its graph is For eample, here is a graph of P() = 5 + 9 + 5 3 5 7 +3 with a 5 = > and odd degree= 5. (d) If a n < and n is odd, then its graph is For eample, here is a graph of P() = 5 9 5 3 + 5 + 7 3 with a 5 = < and odd degree= 5.
Section. A Catalog of Essential Functions Kiryl Tsishchanka 3. Power functions: f() = a where a is a constant. Here we distinguish three main cases: (i) a = n, where n is a positive integer (ii) a = /n, where n is a positive integer (iii) a = 3
Section. A Catalog of Essential Functions Kiryl Tsishchanka. Rational functions: f() = P() Q() where P(), Q() are polynomials. EXAMPLES: f() = +, g() = 3, h() = 3 5+, etc. 3 + 5. Trigonometric functions: In this course it is important to know graphs and basic properties of the following trigonometric functions: sin, cos, tan, cot, sec, csc
Section. A Catalog of Essential Functions Kiryl Tsishchanka 6. Eponential and Logarithmic functions: where a is a positive constant. f() = a, f() = log a IMPORTANT: Do NOT confuse power functions and eponential functions! 5
Section. A Catalog of Essential Functions Kiryl Tsishchanka Transformations of Functions Vertical and Horizontal Shifts: Suppose c >. To obtain the graph of y = f()+c, shift the graph of y = f() a distance c units upward y = f() c, shift the graph of y = f() a distance c units downward y = f( c), shift the graph of y = f() a distance c units to the right y = f(+c), shift the graph of y = f() a distance c units to the left Vertical and Horizontal Stretching and Reflecting: Suppose c >. To obtain the graph of y = cf(), stretch the graph of y = f() vertically by a factor of c y = (/c)f(), compress the graph of y = f() vertically by a factor of c y = f(c), compress the graph of y = f() horizontally by a factor of c y = f(/c), stretch the graph of y = f() horizontally by a factor of c y = f(), reflect the graph of y = f() about the -ais y = f( ), reflect the graph of y = f() about the y-ais EXAMPLES:. Given the graph of f() =, use transformations to graph f() = (+). Step : f() = Step : f() = (+) (horizontal shift) 3 3 - - -3 - -. Given the graph of f() =, use transformations to graph f() =. Step : f() = Step : f() = (vertical shift) 3 - - - - - - 6
Section. A Catalog of Essential Functions Kiryl Tsishchanka 3. Given the graph of f() =, use transformations to graph f() =. Step : f() = Step : f() = (reflection about the -ais) - - 3 - - -3 - - -. Given the graph of f() =, use transformations to graph f() =. Step : f() = Step : f() = (reflection about the y-ais)...8.6.. - -.5 - -.5 5. Given the graph of f() =, use transformations to graph f() = +. Step : f() = Step : f() = + (horizontal shift) Step 3: f() = + (reflection) Step : f() = + (vertical shift) 6. Given the graph of f() =, use transformations to graph f() =. 7
Section. A Catalog of Essential Functions Kiryl Tsishchanka 6. Given the graph of f() =, use transformations to graph f() =. Step : f() = Step : f() = + (horizontal shift) Step 3: f() = + (reflection) Step : f() = + (vertical shift) Step 5: f() = (reflection about the y-ais) 7. Sketch the graph of the function f() = ( 3). 8
Section. A Catalog of Essential Functions Kiryl Tsishchanka Combinations of functions Two functions f and g can be combined to form new functions f +g,f g,fg, and f/g in a manner similar to the way we add, subtract, multiply, and divide real numbers. EXAMPLE: The domain of f() = is A = [, ), the domain of g() = is B = (, ], and the domain of h() = is C = [, ), so the domain of (f g)() = is A B = [,] and (f h)() = is A C = [, ) EXAMPLE: If f() = and g() =, then the domain of the rational function (f/g)() = /( ) is { } or (, ) (, ) There is another way of combining two functions to obtain a new function. For eample, suppose that y = f(u) = u and u = g() = +. Since y is a function of u and u is, in turn, a function of, it follows that y is ultimately a function of. We compute this by substitution: y = f(u) = f(g()) = f( +) = + The procedure is called composition because the new function is composed of the two given functions f and g. EXAMPLE: If f() = + and g() = 3, find the following. (a) f f (b) f g (c) g f (d) g g (e) f(g()) (f) g(f()) 9
Section. A Catalog of Essential Functions Kiryl Tsishchanka EXAMPLE: If f() = + and g() = 3, find the following. (a) f f (b) f g (c) g f (d) g g (e) f(g()) (f) g(f()) Solution: We have (a) f f = ( +) + = + + (b) f g = ( 3) + = 6+ (c) g f = + 3 = (d) g g = 3 3 = 6 (e) f(g()) = ( 3) + = (f) g(f()) = = EXAMPLE: If f() = and g() =, then f f = f g = g f = g g = REMARK: You can see from the Eamples above that sometimes f g = g f, but, in general, f g g f. The domain of f g is the set of all in the domain of g such that g() is in the domain of f. In other words, (f g)() is defined whenever both g() and f(g()) are defined. EXAMPLE: If f() = and g() =, then f f = ( ) = f g =, g f = g g = = (of course, the domain of g g = is all nonnegative numbers). EXAMPLE: If f() = 3 and g() = 3, then f f = ( 3 ) 3 = 9 f g = g f = g g = 3 3 = 9 EXAMPLE: If f() = and g() =, find each function and its domain. (a) f g (b) g f (c) f f (d) g g
Section. A Catalog of Essential Functions Kiryl Tsishchanka EXAMPLE: If f() = and g() =, find each function and its domain. (a) f g (b) g f (c) f f (d) g g Solution: (a) We have (f g)() = f(g()) = f( ) = = The domain of f g is { } = { } = (, ]. (b) We have (g f)() = g(f()) = g( ) = For tobedefinedwemusthave. For tobedefinedwemusthave, that is,, or. Thus we have, so the domain of g f is the closed interval [,]. (c) We have The domain of f f is [, ). (f f)() = f(f()) = f( ) = = (d) We have (g g)() = g(g()) = g( ) = This epression is defined when both and. The first inequality means, and the second is equivalent to, or, or. Thus, so the domain of g g is the closed interval [-, ]. It is possible to take the composition of three or more functions. For instance, the composite function f g h is found by first applying h, then g, and then f as follows: (f g h)() = f(g(h())) EXAMPLE: Find f g h if f() = /(+), g() =, and h() = +3. Solution: We have (f g h)() = f(g(h())) = f(g(+3)) = f((+3) ) = (+3) (+3) + So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following eample. EXAMPLE: Given F() = (+9), find functions f,g, and h such that F = f g h.
Section. A Catalog of Essential Functions Kiryl Tsishchanka EXAMPLE: Given F() = (+9), find functions f,g, and h such that F = f g h. Solution : The formula for F says: First add 9, then square +9, and finally divide by the result. So we let f() =, g() =, h() = +9 Then (f g h)() = f(g(h())) = f(g(+9)) = f((+9) ) = (+9) = F() Solution : Here is an other way to look at F: First add 9, then divide by +9, and finally square the result. So we let Then f() =, g() =, h() = +9 (f g h)() = f(g(h())) = f(g(+9)) = f ( ) ( ) = = +9 +9 (+9) = F()