Combining and Reversing Functions: Quick Revision

A brief note on Combining and Reversing Functions. Read it before a test. Try each question at the end before you open its answer.

Composition

• $(g \circ f)(x) = g(f(x))$: do $f$ first.
• The domain of $g \circ f$ keeps the $x$ for which $f(x)$ can go into $g$.
• Usually $g \circ f \neq f \circ g$.

One-one, onto and inverse

To find $f^{-1}$, solve $y = f(x)$ for $x$. Its graph is the reflection of $f$ in $y = x$, and $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Models

Choose the family from the data: a line for equal steps, a parabola for a square law, a reciprocal for a constant product. Least squares picks the numbers. Trust a model inside its data, not far outside it.

A worked example

$f(x) = 2x - 1$ and $g(x) = x^2 + 1$. Find $g \circ f$, $f \circ g$ and $f^{-1}$.

1. $g \circ f$. $(2x - 1)^2 + 1 = 4x^2 - 4x + 2$.
2. $f \circ g$. $2(x^2 + 1) - 1 = 2x^2 + 1$.
3. $f^{-1}$. $y = 2x - 1$ gives $x = \dfrac{y + 1}{2}$, so $f^{-1}(x) = \dfrac{x + 1}{2}$.

Practice with new numbers

A program to try

The program checks the worked example at three points.

Mistakes to avoid

• Swapping the order of a composite.
• Writing $\dfrac{1}{f(x)}$ for $f^{-1}(x)$.
• Looking for an inverse of a function that is not one-one, such as $x^2$ on $\mathbb{R}$.
• Trusting a model far outside its data.

Check yourself

1. $f(x) = x + 1$ and $g(x) = 2x$. Find $(f \circ g)(4)$.

2. What is the inverse of $f(x) = 7 - x$?

3. Is $f(x) = x^3$ one-one on $\mathbb{R}$?

4. A model $y = \dfrac{k}{x}$ fits the reading $(5, 6)$. What does it give at $x = 3$?