Algebra
Grade 6-9Functions
A function is a rule that takes an input and produces exactly one output, and function notation like f(x) is used to describe this rule and work with it in different ways. This lesson covers evaluating a function, solving an equation of the form f(x) = a, working out a composite function such as fg(x), and finding the inverse of a function.
What you need to know
- f(x) is a rule; f(3) means substitute x = 3 into that rule.
- f g(x) means do g first, then apply f to the result.
- The inverse undoes the function: swap x and y, then rearrange for y.
- The domain is the set of allowed inputs; the range is the set of possible outputs.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Evaluating a function
Substitute the given value in place of x in the function's rule, then work out the result.
For f(x) = 3x − 2, finding f(5) means substituting x = 5: f(5) = 3(5) − 2 = 13.
Solving f(x) = a
Set the function's expression equal to the given value, then solve the resulting equation for x.
For f(x) = 5x − 3, solving f(x) = 17 means solving 5x − 3 = 17, which gives 5x = 20, so x = 4.
Working out a composite function
For fg(x), apply the function g first, then apply f to that result. Work from the inside out, calculating g's output before substituting it into f.
For f(x) = 2x + 1 and g(x) = x − 3, finding fg(2) means first working out g(2) = 2 − 3 = −1, then substituting into f: f(−1) = 2(−1) + 1 = −1.
Finding the inverse of a function
Write the function as y = ..., swap x and y, then rearrange to make y the subject again. The result is the inverse function, f⁻¹(x).
For f(x) = 4x − 3, write y = 4x − 3, swap to get x = 4y − 3, then rearrange: y = (x + 3) ÷ 4, so f⁻¹(x) = (x + 3) ÷ 4.
Worked examples
Three exam-style questions, fully solved
f(x) = 3x − 2. Find f(5).
Easy- 1.Substitute x = 5 into the function: 3(5) − 2
Answer: 13
f(x) = 2x + 1 and g(x) = x − 3. Find fg(2).
Medium- 1.Work out g(2) first: 2 − 3 = −1
- 2.Substitute this result into f: f(−1) = 2(−1) + 1
Answer: −1
f(x) = 3x − 4. Solve ff(x) = 11.
Hard- 1.Find ff(x) by substituting f(x) into f again: ff(x) = 3(3x − 4) − 4
- 2.Simplify: ff(x) = 9x − 16
- 3.Set equal to 11 and solve: 9x − 16 = 11, so 9x = 27
Answer: x = 3
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
f(x) = 3x − 1. Work out f(4).
f(x) = x² + 2. Work out f(−3).
f(x) = 2x + 5. Solve f(x) = 17.
f(x) = 4x and g(x) = x + 3. Work out fg(2).
f(x) = 3x − 6. Find the inverse function f⁻¹(x).
f(x) = x² − 4x. Solve f(x) = 0.
Avoid these
The mistakes students make most often
Making an arithmetic slip with negative numbers when substituting a value into a function, especially when the input itself is negative.
Working out a composite function in the wrong order, such as calculating gf(x) when fg(x) was asked for, or vice versa.
When finding an inverse function, forgetting to swap x and y before rearranging, which leads to the wrong final expression.
When solving f(x) = a, evaluating the function instead of setting its expression equal to a and solving the resulting equation.
Confusing the inverse function, f⁻¹(x), with the reciprocal of the function, 1 ÷ f(x), which are entirely different things despite the similar notation.
FAQ
Questions students and parents ask
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What to learn next
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