ZhixuMathematics | Understanding functions
A function expresses a rule for how one quantity depends on another. Start with the allowed inputs, then examine the outputs. Equations, tables and graphs become three ways to describe the same relationship.
Separate the rule, input and output
In y = f(x), x is the input, f is the rule and y is the output. Every input in the domain must have exactly one output. Different inputs may share an output: for f(x) = x², both 2 and −2 produce 4. This is still a function.
Check the domain before calculating
Over the real numbers, a denominator cannot be zero and a square-root argument cannot be negative. Therefore g(x) = 1/(x−2) requires x ≠ 2, while h(x) = √(x−1) requires x ≥ 1. A real-world context may impose further restrictions, such as non-negative elapsed time.
Read a linear function as a rate and an intercept
For f(x) = 2x + 3, increasing x by 1 increases the output by 2. At x = 0, the output is 3. The slope 2 is the rate of change; 3 is the vertical intercept. Inputs −1, 0 and 1 give outputs 1, 3 and 5. Plotting these points provides a visual check.
Distinguish evaluation from solving
Evaluating f(4) means substituting a known input: 2 × 4 + 3 = 11. Solving f(x) = 11 means finding an input from a known output: 2x + 3 = 11 gives x = 4. A nonlinear function may have several inputs for one output: x² = 4 has real solutions 2 and −2.
Verify using another representation
Substitute your solution into the original equation and check that it lies in the domain. On a graph, label the axes and units. Do not assume a curve is increasing everywhere simply because part of it rises: x² behaves differently on the negative and positive half-axes.
Try it yourself
Let f(x) = 3x − 2. Find f(5), then find x when f(x) = 10.
Show reasoning and answer
f(5) = 3 × 5 − 2 = 13. Solving 3x − 2 = 10 gives x = 4; substitution returns 10. The first question goes from input to output, and the second reverses the process.
Use the method in your next study session
Make a table with input, rule and output columns. Represent the same function with values, an equation and a graph, and practise translating between them.
These guides explain general subject foundations for learning and revision. They are not official examination syllabuses or questions. Refer to the relevant official institutions for examination requirements.