Skip to main content
Mathovia

Math Practice

Function Notation Practice Problems

Fourteen hand-built problems on function notation, each fully worked. They cover evaluating at positive and negative numbers, substituting a whole expression, solving f(x) = b backwards, the difference between changing the input and changing the output, and reading function notation in a real model.

  1. 1.

    For \(f(x) = 4x - 1\), find \(f(3)\).

    Show full solution

    \(f(3) = 4(3) - 1 = 12 - 1 = 11\).

  2. 2.

    For \(f(x) = 4x - 1\), find \(f(-2)\).

    Show full solution

    \(f(-2) = 4(-2) - 1 = -8 - 1 = -9\).

  3. 3.

    For \(f(x) = 3x - 5\), find \(f(0)\).

    Show full solution

    \(f(0) = 3(0) - 5 = -5\). Evaluating at zero always gives the \(y\)-intercept.

  4. 4.

    For \(g(x) = x^2 + 5\), find \(g(-3)\).

    Show full solution

    \(g(-3) = (-3)^2 + 5 = 9 + 5 = 14\). The brackets matter: \((-3)^2\) is \(+9\), not \(-9\).

  5. 5.

    For \(h(x) = \dfrac{12}{x}\), find \(h(4)\).

    Show full solution

    \(h(4) = \dfrac{12}{4} = 3\).

  6. 6.

    For \(h(x) = \dfrac{12}{x}\), find \(h(-6)\).

    Show full solution

    \(h(-6) = \dfrac{12}{-6} = -2\).

  7. 7.

    For \(g(x) = x^2 - 4x\), find \(g(-3)\).

    Show full solution

    \(g(-3) = (-3)^2 - 4(-3) = 9 + 12 = 21\). Both signs matter — the square gives \(+9\) and subtracting a negative gives \(+12\).

  8. 8.

    For \(f(x) = 5x + 2\), solve \(f(x) = 27\).

    Show full solution

    \(5x + 2 = 27\), so \(5x = 25\) and \(x = 5\). This runs the function backwards — it is just solving an equation.

  9. 9.

    For \(f(x) = x^2\), write \(f(2t)\) in simplest form.

    Show full solution

    \(f(2t) = (2t)^2 = 4t^2\). The whole input gets squared, not just the \(t\).

  10. 10.

    For \(f(x) = x^2 - 1\), write \(f(a + 1)\) expanded, in terms of \(a\).

    Show full solution

    \(f(a+1) = (a+1)^2 - 1 = a^2 + 2a + 1 - 1 = a^2 + 2a\).

  11. 11.

    For \(f(x) = 3x\), write \(f(x + 2)\) expanded.

    Show full solution

    \(f(x+2) = 3(x+2) = 3x + 6\). Compare with \(f(x) + 2 = 3x + 2\) — changing the input is not the same as changing the output.

  12. 12.

    For \(f(x) = x^2 - 9\), solve \(f(x) = 0\). Give both values.

    Show full solution

    \(x^2 - 9 = 0\) gives \(x^2 = 9\), so \(x = 3\) or \(x = -3\). These are the zeros of the function, and the \(x\)-intercepts of its graph.

  13. 13.

    For \(f(x) = 2x + 1\) and \(g(x) = x^2\), find \(f(g(3))\).

    Show full solution

    Inner first: \(g(3) = 9\). Then \(f(9) = 2(9) + 1 = 19\).

  14. 14.

    A taxi charges \(C(m) = 1.5m + 3\) pounds for \(m\) miles. How many miles cost \(\pounds 24\)?

    Show full solution

    \(1.5m + 3 = 24\), so \(1.5m = 21\) and \(m = 14\) miles. The \(3\) is the flag-fall and the \(1.5\) is the per-mile rate.
M
Written by
Mathovia Team
Editorial Team

Function Notation Practice Problems

The Function Notation lesson covers how to read and why it beats plain ; this set is fourteen problems to practise on, each fully solved.

How to use this set

The whole skill is one instruction: replace every in the rule with whatever is inside the brackets. Do it with the brackets still in place, then simplify.

Most problems want a single number. A few want an expression in another letter — type it as you would write it, for example 4t^2.

Common mistakes to watch for

Reading as multiplication. The brackets hold an input.

Dropping brackets on a negative input. with gives .

Substituting into only some of the s. Every occurrence gets replaced.

Confusing with . The first doubles the input, the second doubles the output.

Expecting one answer to . There can be two, or none — here has two.

Where to go next

Function Operations combines and chains functions, Domain and Range asks what may go in and what comes out, and Piecewise Functions evaluates rules that change partway along.

Frequently Asked Questions

What does f(x) mean?+

It is read "f of x" and means the output of the function when the input is . The brackets hold the input — is not multiplied by .

How do you evaluate a function?+

Replace every in the rule with whatever is inside the brackets, then simplify. Keep the brackets while you substitute, especially for a negative input.

Why does f(-3) need brackets?+

Because but . Writing the substitution with brackets first and simplifying second removes the ambiguity.

What is the difference between f(x+1) and f(x)+1?+

changes the input before the rule runs; runs the rule and then changes the output. For they give and .

What does solving f(x) = 0 find?+

The inputs that produce an output of zero — the zeros or roots of the function, which are the -intercepts of its graph. There may be none, one, or several.

Is f⁻¹(x) the same as 1 over f(x)?+

No. is the inverse function; is the reciprocal. The is notation, not an exponent.

More practice problems

Function Operations and Composition Practice Problems Domain and Range Practice Problems Piecewise Functions Practice Problems