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) = 4x + 9 and g(x) = (x-9)/4

Verified step by step guidance

1. First, find the composition \( f(g(x)) \). This means you substitute \( g(x) \) into \( f(x) \). So, replace every \( x \) in \( f(x) = 4x + 9 \) with \( g(x) = \frac{x - 9}{4} \). The expression becomes \( f\left(g(x)\right) = 4 \left( \frac{x - 9}{4} \right) + 9 \).
2. Next, simplify the expression for \( f(g(x)) \). Multiply 4 by \( \frac{x - 9}{4} \) and then add 9. This will help you see if the composition simplifies to \( x \).
3. Now, find the composition \( g(f(x)) \). This means you substitute \( f(x) \) into \( g(x) \). Replace every \( x \) in \( g(x) = \frac{x - 9}{4} \) with \( f(x) = 4x + 9 \). The expression becomes \( g\left(f(x)\right) = \frac{(4x + 9) - 9}{4} \).
4. Simplify the expression for \( g(f(x)) \). Subtract 9 from \( 4x + 9 \) and then divide by 4. Check if this simplifies to \( x \).
5. Finally, determine if \( f \) and \( g \) are inverses by checking if both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both do, then \( f \) and \( g \) are inverse functions of each other.

Key Concepts

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, solving equations, and substituting variables accurately. It is essential for correctly computing compositions and verifying inverse relationships by simplifying the composed functions to see if they equal x.