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

Verified step by step guidance

1. Step 1: To find $f(g(x))$, substitute $g(x)=\frac{x-9}{4}$ into $f(x)=4x+9$. This means replacing $x$ in $f(x)$ with $\frac{x-9}{4}$.
2. Step 2: Simplify the expression $f\left(\frac{x-9}{4}\right)=4\left(\frac{x-9}{4}\right)+9$.
3. Step 3: To find $g(f(x))$, substitute $f(x)=4x+9$ into $g(x)=\frac{x-9}{4}$. This means replacing $x$ in $g(x)$ with $4x+9$.
4. Step 4: Simplify the expression $g(4x+9)=\frac{(4x+9)-9}{4}$.
5. Step 5: Determine if $f(g(x))=x$ and $g(f(x))=x$. If both are true, then $f$ 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, 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) takes an input x and produces an output y, then g(y) should return the input x. To determine if f and g are inverses, we need to check if f(g(x)) = x and g(f(x)) = x for all x in their domains. This concept is essential for verifying the relationship between the given functions.

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 f(x) = 4x + 9 and g(x) = (x-9)/4 are both linear, which means their graphs are straight lines. Recognizing their linearity helps in understanding their behavior and the nature of their inverses, as linear functions have well-defined slopes and intercepts.