Given f(x) = (x + 5)/4, find f−1(x).
- Write y = (x + 5)/4, then swap x and y: x = (y + 5)/4.
- Multiply by 4: 4x = y + 5, so y = 4x − 5.
- Check: f(3) = (3 + 5)/4 = 2 and 4 × 2 − 5 = 3, so the inverse returns the input.
Answer: f−1(x) = 4x − 5
Functions, graphs and their key features
Maths: Applications and Interpretation · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
A function is a rule that gives exactly one output for each input. The domain is the set of inputs allowed and the range is the set of outputs that result. In a real model the domain is limited by the context, for example time from 0 until a ball lands.
The inverse function f−1 undoes f: if f sends 3 to 2, then f−1 sends 2 back to 3. Its graph is the graph of f reflected in the line y = x.
Use your graphing calculator to draw the graph and read its key features: intercepts, zeros, maximum and minimum points, the vertex, lines of symmetry, asymptotes, and points where two graphs cross.
Rules to remember
- f(a) means put x = a into the rule. Use brackets for negatives: (−2)2 = 4
- To find f−1(x): write y = f(x), swap x and y, then make y the subject
- The domain of f−1 is the range of f, and the range of f−1 is the domain of f
- Zeros (x-intercepts): solve f(x) = 0. y-intercept: work out f(0)
- Quadratic y = ax2 + bx + c: axis of symmetry and vertex at x = −b/(2a)
- y = a + b/(x − c) has vertical asymptote x = c and horizontal asymptote y = a
- For the range on a restricted domain, check the end points and any turning point inside the domain
Common mistake
Students read f−1(x) as 1/f(x). It is the inverse function, not the reciprocal. For the range, they also test only the end points of the domain and miss a minimum or maximum that lies between them.
Worked example
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