IGCSE Maths / Functions and function notation
Solving equations written in function notation revision notes
Edexcel International GCSE Maths (4MA1), chapter 24: Functions and function notation, specification point 3.2b. Free lesson.
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To solve an equation such as f(x) = 11, replace f(x) with its rule and solve the resulting equation for x. Remember that if x is squared, both the positive and negative square roots may work.
Turn function notation into an equation
In a function question, the number on the right is the output. The rule uses the input, x, to produce that output. Replace f(x) with the rule, then solve the equation like any other equation. Put the given output on the right of the equals sign. Check which value of x makes the rule give that output.
Solve a linear function equation
For f(x) = 3x − 4 and f(x) = 11, the rule must equal 11. Use inverse operations on both sides to leave x by itself. Adding 4 undoes subtracting 4. Dividing by 3 undoes multiplying by 3.
Keep the input and output in the right places
A common error is to put 11 into the rule, as if it were the input. That calculates f(11), which is a different question. In f(x) = 11, the 11 is the output, so it goes on the right of the equation. 3 × 11 − 4 = 29, but this is f(11), not the solution to f(x) = 11. Solve 3x − 4 = 11 to find the input that gives an output of 11.
A squared x can give two answers
If the rule includes x², isolate x² first. Taking the square root undoes squaring, but both 5 and −5 square to 25. Write both answers using ±. The square root of 25 is 5, but (−5)² is also 25. Check both values in the original rule.
Use the same rule throughout a question
A question may ask you to evaluate a function, solve for an input, or solve an equation with x on both sides. In each part, use the stated rule and then follow what the question asks. For f(x) = 5x + 2, f(3) = 5 × 3 + 2 = 17. To solve f(x) = 32, write 5x + 2 = 32, so 5x = 30 and x = 6. To solve f(x) = 2x + 11, write 5x + 2 = 2x + 11, so 3x = 9 and x = 3.
Check the answer by putting it back in
Substitute your answer for x in the original rule. The output should match the number in the question. This can catch a slip and confirms that both answers work when x is squared. For g(x) = 2x² + 1 and g(x) = 51, subtract 1 to get 2x² = 50, then divide by 2 to get x² = 25. Therefore x = ±5. Both give 2 × 25 + 1 = 51.