Inverse Functions

For simple linear functions, you can think of a function machine where you "reverse the arrows" and use opposite operations. For example, if $f(x) = 3x - 5$, the forward process is $\times 3$ then $-5$. The reverse process is $+5$ then $\div 3$, giving an algebraic result of $f^{-1}(x) = \frac{x + 5}{3}$.

Finding Inverse Functions Algebraically

While function machines work for simple equations, harder exam questions require a formal algebraic method. To find $f^{-1}(x)$, the original function $f(x)$ must be rearranged to make $x$ the subject of a formula.

A highly recommended method to reduce algebraic errors is the "swap" method. First, replace $f(x)$ with $y$, then swap the $x$ and $y$ variables entirely before rearranging the equation to isolate the new $y$.

Find the inverse of $f(x) = 2x + 1$.

Step 1: Write the function as an equation using $y$.

• $y = 2x + 1$

Step 2: Swap the $x$ and $y$ variables.

• $x = 2y + 1$

Step 3: Rearrange to isolate $y$.

• $x - 1 = 2y$
• $y = \frac{x - 1}{2}$

Step 4: State the final answer using inverse notation.

• $f^{-1}(x) = \frac{x - 1}{2}$

Inverses with Algebraic Fractions

Questions where the variable appears twice, such as in algebraic fractions, are common at Higher Tier. These require expanding brackets and factorising to successfully change the subject.

Find $f^{-1}(x)$ for $f(x) = \frac{x+5}{x+3}$.

Step 1: Set the expression to $y$ and swap the variables.

• $x = \frac{y+5}{y+3}$

Step 2: Multiply by the denominator to clear the fraction.

• $x(y+3) = y+5$

Step 3: Expand the brackets.

• $xy + 3x = y + 5$

Step 4: Collect all terms containing $y$ on one side of the equation.

• $xy - y = 5 - 3x$

Step 5: Factorise out $y$.

• $y(x - 1) = 5 - 3x$

Step 6: Divide to isolate $y$ and write the final inverse function.

• $y = \frac{5 - 3x}{x - 1}$
• $f^{-1}(x) = \frac{5 - 3x}{x - 1}$

Domain and Range of Inverse Functions

Why do some functions lack an inverse unless we strictly limit the input numbers? For an inverse function to exist, the original function must be a one-to-one mapping, meaning each input maps to a unique output.

Many-to-one functions, such as $f(x) = x^2$, require a restricted domain (e.g., $x \ge 0$) to become one-to-one so that a unique inverse can be defined. You must also exclude values that result in an undefined output, such as values that cause division by zero or negative square roots.

The most important rule connecting a function and its inverse is that their inputs and outputs are directly swapped. The domain of $f^{-1}(x)$ is exactly the range of $f(x)$, and the range of $f^{-1}(x)$ is exactly the domain of .

Graphs of Inverse Functions

When plotted on a coordinate grid, the graphical relationship between a function and its inverse is a visual reflection. The graph of $y = f^{-1}(x)$ is always a reflection of the graph $y = f(x)$ in the diagonal line $y=$.

Because the $x$ and $y$ values swap, any coordinate point $(a, b)$ that lies on the graph of $f(x)$ will transform into the point $(b, a)$ on the graph of . For example, if crosses the -axis at , must cross the -axis at .