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)=5x-9 and g(x) = (x+5)/9

Verified step by step guidance

1. Step 1: To find \( f(g(x)) \), substitute \( g(x) = \frac{x+5}{9} \) into \( f(x) = 5x - 9 \). This gives \( f(g(x)) = 5\left(\frac{x+5}{9}\right) - 9 \).
2. Step 2: Simplify \( f(g(x)) \) by distributing the 5: \( f(g(x)) = \frac{5(x+5)}{9} - 9 \).
3. Step 3: Simplify further by expanding and combining terms: \( f(g(x)) = \frac{5x + 25}{9} - 9 \).
4. Step 4: To find \( g(f(x)) \), substitute \( f(x) = 5x - 9 \) into \( g(x) = \frac{x+5}{9} \). This gives \( g(f(x)) = \frac{(5x-9)+5}{9} \).
5. Step 5: Simplify \( g(f(x)) \) by combining like terms: \( g(f(x)) = \frac{5x - 4}{9} \).

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, the composition f(g(x)) means you first apply g to x, 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. To determine if f and g are inverses, one must check if f(g(x)) = x and g(f(x)) = x, which is essential for the problem at hand.

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) = 5x - 9 and g(x) = (x + 5)/9, are both linear. Understanding their properties, such as how to manipulate and graph them, is important for performing the necessary calculations and verifying their inverse relationship.