16. Function Composition, Inverses, and Transformations

Function notation becomes challenging when several operations are layered. Read from the inside out and keep each substitution in parentheses.

Example 1 - composition:
\[ f(x)=2x-3,\qquad g(x)=x^2+1. \]
Then
\[ f(g(x))=2(x^2+1)-3=2x^2-1, \]
while
\[ g(f(x))=(2x-3)^2+1=4x^2-12x+10. \]
Composition order matters.

To find an inverse, replace \(f(x)\) with \(y\), switch \(x\) and \(y\), then solve for \(y\). For \(f(x)=3x+6\):
\[ y=3x+6, \]
switch to \(x=3y+6\), then solve:
\[ f^{-1}(x)=\frac{x-6}{3}. \]
Check by composing: \(f(f^{-1}(x))=x\).

Transformations describe how a parent graph moves:

• \(f(x)+k\): up \(k\).
• \(f(x-h)\): right \(h\).
• \(-f(x)\): reflection across the \(x\)-axis.
• \(f(-x)\): reflection across the \(y\)-axis.

The horizontal sign feels backward. For \(y=(x-4)^2\), the vertex moves right from \((0,0)\) to \((4,0)\).

When a table defines a function, each input may have only one output. Repeated outputs are allowed; repeated inputs paired with different outputs are not.

Common trap: treating \(f^{-1}(x)\) as \(1/f(x)\). An inverse function reverses the input-output process; it is not a reciprocal.

Advanced mastery lab

Learning targets

• compose functions in the correct order
• construct and verify an inverse function
• interpret horizontal and vertical transformations accurately

Visual model

How to read this SVG. Composition connects machines in sequence; an inverse reverses every step and its order. Verify that forward-then-reverse returns the original input.

Decision protocol

1. For composition, start with the innermost input.
2. For an inverse, write \(y=f(x)\), swap \(x\) and \(y\), then solve for \(y\).
3. Verify inverses by composition.
4. Separate inside input changes from outside output changes.

Decision table

Timed transfer challenge: Build and verify an inverse

Target time. 4 minutes, including a composition check.

Problem. Find the inverse of \(f(x)=\frac{x+4}{3}\).

Solution.

1. Write \(y=(x+4)/3\), then swap: \(x=(y+4)/3\).
2. Multiply by 3: \(3x=y+4\).
3. Subtract 4: \(f^{-1}(x)=3x-4\).
4. Check: \(f(3x-4)=x\).

Answer. \(f^{-1}(x)=3x-4\)

Second independent check

Retrieval and reflection

Explain the difference between an inverse function and a reciprocal.

Mastery standard. Complete the work accurately, explain the decisive step, and verify the answer using the original condition or an independent representation.