Algebra
Relations and functions
Function notation
20 practice questions
2 video lessons
Theory + worked examples
Theory
Function notation \(f(x)\) (“\(f\) of \(x\)”) names the output for the input \(x\):
\[f(x)=y.\]
\(f(3)\) means the output when \(x=3\).
\(f(x)\) is not multiplication — it names the function's value.
What \(f(x)\) means.
Evaluating \(f(x)=2x+3\).
Evaluating:
\[f(a)=\text{the value of } f \text{ when } x=a\]
Substitute the input everywhere \(x\) appears.
How to use function notation
- To evaluate \(f(a)\), substitute \(a\) for \(x\).
- Simplify to get the output.
- To find an input, set \(f(x)\) equal and solve.
- Interpret in context if given.
Example 1 — Evaluate
For \(f(x)=2x+3\), find \(f(4)\).
Solution
Substitute \(x=4\).
| \(f(4)\) | \(=\) | \(2(4)+3\) |
| \(=\) | \(11\) |
Example 2 — A square
For \(f(x)=x^2\), find \(f(-3)\).
Solution
Square the input.
| \(f(-3)\) | \(=\) | \((-3)^2=9\) |
Example 3 — Solve for the input
For \(f(x)=2x+3\), find \(x\) when \(f(x)=10\).
Solution
Set the output equal and solve.
| \(2x+3\) | \(=\) | \(10\) |
| \(x\) | \(=\) | \(3.5\) |
Example 4 — Interpret
If \(C(h)=15h\) gives cost for \(h\) hours, interpret \(C(2)=30\).
Solution
\(2\) hours cost \(\$30\).
Common pitfalls
\(f(x)\) is not \(f\times x\).
Substitute the input for every \(x\).
\(f(a)\) is the output, not the input.
Frequently asked questions
What does \(f(x)\) mean?
The output of function \(f\) for the input \(x\).
How do you find \(f(4)\)?
Substitute \(4\) for \(x\) and simplify.
Is \(f(x)\) multiplication?
No — it is function notation.
What does \(f(2)=5\) tell you?
When the input is \(2\), the output is \(5\).
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