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 and g(x) = x/4

Verified step by step guidance

1. Step 1: To find $f(g(x))$, substitute $g(x)=\frac{x}{4}$ into $f(x)=4x$. This gives $f(g(x))=4\left(\frac{x}{4}\right)$.
2. Step 2: Simplify $f(g(x))=4\left(\frac{x}{4}\right)$ to get $f(g(x))=x$.
3. Step 3: To find $g(f(x))$, substitute $f(x)=4x$ into $g(x)=\frac{x}{4}$. This gives $g(f(x))=\frac{4x}{4}$.
4. Step 4: Simplify $g(f(x))=\frac{4x}{4}$ to get $g(f(x))=x$.
5. Step 5: 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 functions f and g, the composition f(g(x)) means applying g first and then f to the result. This process is essential for evaluating how two functions interact and can reveal properties such as inverses.

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 original input x. To determine if two functions are inverses, one must check if f(g(x)) = x and g(f(x)) = x for all x in their domains.

Linear Functions

Linear functions are polynomial functions of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this case, f(x) = 4x and g(x) = x/4 are both linear functions, which simplifies the process of finding their compositions and checking for inverses due to their predictable behavior and straightforward algebraic manipulation.