In Exercises 1-10, 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

1. Step 1: To find \( f(g(x)) \), substitute \( g(x) = \frac{x-8}{3} \) into \( f(x) = 3x + 8 \). This gives \( f(g(x)) = 3\left(\frac{x-8}{3}\right) + 8 \).
2. Step 2: Simplify \( f(g(x)) = 3\left(\frac{x-8}{3}\right) + 8 \) by distributing the 3, which results in \( x - 8 + 8 \).
3. Step 3: Simplify further to get \( f(g(x)) = x \).
4. Step 4: To find \( g(f(x)) \), substitute \( f(x) = 3x + 8 \) into \( g(x) = \frac{x-8}{3} \). This gives \( g(f(x)) = \frac{(3x+8)-8}{3} \).
5. Step 5: Simplify \( g(f(x)) = \frac{3x}{3} \) to get \( g(f(x)) = x \). Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are inverses of each other.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For example, if f(x) and g(x) are two functions, then f(g(x)) means you first apply g to x, and then apply f to the result. Understanding this concept is crucial for solving the problem as it requires calculating both f(g(x)) and g(f(x)).

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f. In this question, determining whether f and g are inverses involves checking if f(g(x)) = x and g(f(x)) = x, which is essential for confirming their relationship.

Linear Functions

Linear functions are mathematical expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. The functions given in the problem, f(x) = 3x + 8 and g(x) = (x - 8)/3, are both linear. Understanding their properties, such as how to manipulate and graph them, is important for performing the necessary calculations and analyzing their inverses.