Problem Set Sample Solutions Problems 5 7 serve to review the process of computing f(g(x)) for given functions f and g in preparation for work with inverses of functions in Lesson 19. 1. Sketch the graphs of the functions f(x) = 5 x and g(x) = log 5 (x).. Sketch the graphs of the functions f(x) = ( 1 ) x and g(x) = log1(x). 93 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II--TE-1.3.0-08.015
3. Sketch the graphs of the functions f 1 (x) = ( 1 )x and f (x) = ( 3 )x on the same sheet of graph paper, and answer the following questions. a. Where do the two exponential graphs intersect? The graphs intersect at the point (0, 1). b. For which values of x is ( 1 )x < ( 3 )x? If x > 0, then ( 1 )x < ( 3 )x. c. For which values of x is ( 1 )x > ( 3 )x? If x < 0, then ( 1 )x > ( 3 )x. d. What happens to the values of the functions f 1 and f as x? As x, both f 1 (x) 0 and f (x) 0. e. What are the domains of the two functions f 1 and f? Both functions have domain (, ). 9 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II--TE-1.3.0-08.015
. Use the information from Problem 3 together with the relationship between graphs of exponential and logarithmic functions to sketch the graphs of the functions g 1 (x) = log1(x) and g (x) = log3(x) on the same sheet of graph paper. Then, answer the following questions. a. Where do the two logarithmic graphs intersect? The graphs intersect at the point (1, 0). b. For which values of x is log1(x) < log3(x)? When x < 1, we have log1(x) < log3(x). c. For which values of x is log1(x) > log3(x)? When x > 1, we have log1(x) > log3(x). d. What happens to the values of the functions g 1 and g as x? As x, both g 1 (x) and g (x). e. What are the domains of the two functions g 1 and g? Both functions have domain (0, ). 95 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II--TE-1.3.0-08.015
5. For each function f, find a formula for the function h in terms of x. a. If f(x) = x 3, find h(x) = 18f ( 1 x) + f(x). h(x) = 10x 3 b. If (x) = x + 1, find h(x) = f(x + ) f(). h(x) = x + x c. If f(x) = x 3 + x + 5x + 1, find h(x) = h(x) = x + 1 f(x) + f( x). d. If f(x) = x 3 + x + 5x + 1, find h(x) = h(x) = x 3 + 5x f(x) f( x). 6. In Problem 5, parts (c) and (d), list at least two aspects about the formulas you found as they relate to the function f(x) = x 3 + x + 5x + 1. The formula for 1(c) is all of the even power terms of f. The formula for 1(d) is all of the odd power terms of f. The sum of the two functions gives f back again; that is, f(x)+f( x) + f(x) f( x) = f. 7. For each of the functions f and g below, write an expression for (i) f(g(x)), (ii) g(f(x)), and (iii) f(f(x)) in terms of x. a. f(x) = x 3, g(x) = x 1 8 g(f(x)) = x 8 i f(f(x)) = x 9 b. f(x) = b x a, g(x) = b + a for two numbers a and b, when x is not 0 or a x g(f(x)) = x b i f(f(x)) = b, which is equivalent to f(f(x)) = b(x a) a b+a ax x a c. f(x) = x+1 x+1, g(x) =, when x is not 1 or 1 x 1 x 1 i g(f(x)) = x f(f(x)) = x d. f(x) = x, g(x) = log (x) i f(f(x)) = x 96 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II--TE-1.3.0-08.015
e. f(x) = ln(x), g(x) = e x i f(f(x)) = ln(ln(x)) f. f(x) = 100 x, g(x) = 1 log (1 x) i f(f(x)) = 10000 100x 97 This work is derived from Eureka Math and licensed by Great Minds. 015 Great Minds. eureka-math.org This file derived from ALG II--TE-1.3.0-08.015