Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 3

Chapter 3, Problem 3

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

Verified step by step guidance

First, find the composition \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \). This means replacing every \( x \) in \( f(x) = 3x + 8 \) with \( g(x) = \frac{x - 8}{3} \). So, write \( f(g(x)) = 3 \left( \frac{x - 8}{3} \right) + 8 \).

Next, simplify the expression for \( f(g(x)) \) by performing the multiplication and addition inside the function. Multiply 3 by \( \frac{x - 8}{3} \) and then add 8.

Then, find the composition \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \). Replace every \( x \) in \( g(x) = \frac{x - 8}{3} \) with \( f(x) = 3x + 8 \). So, write \( g(f(x)) = \frac{(3x + 8) - 8}{3} \).

Simplify the expression for \( g(f(x)) \) by performing the subtraction in the numerator and then dividing by 3.

Finally, determine whether \( f \) and \( g \) are inverses by checking if both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both equal \( x \), then \( f \) and \( g \) are inverse functions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to combine functions and analyze their combined effect.

Inverse Functions

Inverse functions reverse the effect of each other, meaning f(g(x)) = x and g(f(x)) = x for all x in the domain. To verify if two functions are inverses, we check if their compositions yield the identity function, which returns the input unchanged.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions, substituting variables, and solving equations. It is essential for correctly performing function composition and verifying inverse relationships by simplifying the composed functions to check if they equal x.

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