Working with Composite and Inverse Functions

Composite Functions: Chaining Rules

A composite function applies one function to the result of another. The notation $fg(x)$ means apply $g$ first, then apply $f$ to the output.

Inverse Functions: Undoing the Rule

An inverse function, written $f^{-1}(x)$, reverses the original function. If $f(x)$ takes you from A to B, then $f^{-1}(x)$ takes you back from B to A.

Key facts:

• $f^{-1}(f(x)) = x$.
• The graph of $f^{-1}(x)$ is a reflection of $f(x)$ in the line $y = x$.
• Important: $f^{-1}(x)$ does not mean $\dfrac{1}{f(x)}$.

Example 1: Evaluating Composite Functions

Given $f(x) = 3x – 2$ and $g(x) = x^2 + 1$, find $fg(2)$.

1. Work inside out — find $g(2)$ first.
   $g(2) = (2)^2 + 1 = 5$.
2. Use this as the input for $f$: find $f(5)$.
   $f(5) = 3(5) – 2 = 13$.

Answer: $fg(2) = 13$

Example 2: Finding a Composite Expression

Given $f(x) = 2x + 5$ and $g(x) = x – 3$, find an expression for $fg(x)$.

1. $fg(x) = f(g(x))$ — substitute $g(x)$ into $f$ in place of $x$.
2. $f(g(x)) = 2(x – 3) + 5$.
3. Expand and simplify: $2x – 6 + 5$.

Answer: $fg(x) = 2x – 1$

Example 3: Finding an Inverse Function

Find the inverse of $f(x) = 4x + 7$.

1. Replace $f(x)$ with $y$: $y = 4x + 7$.
2. Rearrange to make $x$ the subject:
   $y – 7 = 4x \implies x = \dfrac{y-7}{4}$.
3. Swap $x$ and $y$: $y = \dfrac{x-7}{4}$.
4. Replace $y$ with $f^{-1}(x)$.

Answer: $f^{-1}(x) = \dfrac{x-7}{4}$

Example 4: Inverse with a Square Root

Find the inverse of $g(x) = \sqrt{x – 3}$ for $x \geq 3$.

1. Let $y = \sqrt{x – 3}$.
2. Square both sides: $y^2 = x – 3$.
3. Make $x$ the subject: $x = y^2 + 3$.
4. Swap $x$ and $y$: $y = x^2 + 3$.

Answer: $g^{-1}(x) = x^2 + 3$

🤔 Common Misunderstandings

• Order of operations: Mixing up $fg(x)$ and $gf(x)$. Work from right to left — apply $g$ first, then $f$.
• Inverse notation: Thinking $f^{-1}(x)$ means $\frac{1}{f(x)}$. The $^{-1}$ means inverse, not a reciprocal.
• Forgetting to swap: Missing the final step of swapping $x$ and $y$ when finding an inverse.

📝 Common Exam Mistakes

• Algebraic errors when expanding brackets or rearranging during the inverse method.
• Incorrect order: Calculating $gf(x)$ when $fg(x)$ was asked. Read carefully!
• Not showing working: All steps of rearrangement must be visible to earn full method marks for inverse functions.