Function Evaluator Notes

Published on: September 29, 2026

This Function Evaluator Calculator works out what a function returns for one input value, and shows the working line by line. Every x in the rule stands for the same input, so each one is replaced by that value before anything is calculated. Each term is then worked out on its own, which keeps the signs and the powers from getting tangled, and only afterwards are the terms added together. Following the same pattern every time makes f of a value easy to read off.

Step-by-step method

  1. Set up the problem.
  2. Replace every x in the function rule with the input value.
  3. Work out each term on its own.
  4. Add and subtract the terms to get the output.
See Example 1 Hide Example 1

Problem

\(f(x) = x^{3} - 2x + 1,\; x = -2\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Replace every x in the function rule with the input value.
  3. Work out each term on its own.
  4. Add and subtract the terms to get the output.

Step 1

Step 1 - Set up the problem.

In this problem: The function rule is \(f(x) = x^{3} - 2x + 1\) and the input is \(x = -2\).

\(f(x) = x^{3} - 2x + 1,\; x = -2\)

Step 2

Step 2 - Replace every x in the function rule with the input value.

In this problem: Every \(x\) in the rule stands for the input, so each one becomes \(\left(-2\right)\), giving \(f\left(-2\right) = \left(-2\right)^{3} - 2\left(-2\right) + 1\).

\(f\left(-2\right) = \left(-2\right)^{3} - 2\left(-2\right) + 1\)

Step 3

Step 3 - Work out each term on its own.

In this problem: Working out each term on its own turns the expression into \(-8 + 4 + 1\).

\(f\left(-2\right) = -8 + 4 + 1\)

Step 4

Step 4 - Add and subtract the terms to get the output.

In this problem: Adding and subtracting \(-8 + 4 + 1\) leaves \(-3\).

\(f\left(-2\right) = -3\)

Final Answer

\(f\left(-2\right) = -3\)
See Example 2 Hide Example 2

Problem

\(f(x) = x^{2} + 2x + 1,\; x = \frac{1}{2}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Replace every x in the function rule with the input value.
  3. Work out each term on its own.
  4. Add and subtract the terms to get the output.

Step 1

Step 1 - Set up the problem.

In this problem: The function rule is \(f(x) = x^{2} + 2x + 1\) and the input is \(x = \frac{1}{2}\).

\(f(x) = x^{2} + 2x + 1,\; x = \frac{1}{2}\)

Step 2

Step 2 - Replace every x in the function rule with the input value.

In this problem: Every \(x\) in the rule stands for the input, so each one becomes \(\left(\frac{1}{2}\right)\), giving \(f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^{2} + 2\left(\frac{1}{2}\right) + 1\).

\(f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^{2} + 2\left(\frac{1}{2}\right) + 1\)

Step 3

Step 3 - Work out each term on its own.

In this problem: Working out each term on its own turns the expression into \(\frac{1}{4} + 1 + 1\).

\(f\left(\frac{1}{2}\right) = \frac{1}{4} + 1 + 1\)

Step 4

Step 4 - Add and subtract the terms to get the output.

In this problem: Adding and subtracting \(\frac{1}{4} + 1 + 1\) leaves \(\frac{9}{4}\).

\(f\left(\frac{1}{2}\right) = \frac{9}{4}\)

Final Answer

\(f\left(\frac{1}{2}\right) = \frac{9}{4}\)
See Example 3 Hide Example 3

Problem

\(f(x) = 3x^{2} - 4x + 2,\; x = 2\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Replace every x in the function rule with the input value.
  3. Work out each term on its own.
  4. Add and subtract the terms to get the output.

Step 1

Step 1 - Set up the problem.

In this problem: The function rule is \(f(x) = 3x^{2} - 4x + 2\) and the input is \(x = 2\).

\(f(x) = 3x^{2} - 4x + 2,\; x = 2\)

Step 2

Step 2 - Replace every x in the function rule with the input value.

In this problem: Every \(x\) in the rule stands for the input, so each one becomes \(\left(2\right)\), giving \(f\left(2\right) = 3\left(2\right)^{2} - 4\left(2\right) + 2\).

\(f\left(2\right) = 3\left(2\right)^{2} - 4\left(2\right) + 2\)

Step 3

Step 3 - Work out each term on its own.

In this problem: Working out each term on its own turns the expression into \(12 - 8 + 2\).

\(f\left(2\right) = 12 - 8 + 2\)

Step 4

Step 4 - Add and subtract the terms to get the output.

In this problem: Adding and subtracting \(12 - 8 + 2\) leaves \(6\).

\(f\left(2\right) = 6\)

Final Answer

\(f\left(2\right) = 6\)
See Example 4 Hide Example 4

Problem

\(f(x) = 2x - 5,\; x = -3\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Replace every x in the function rule with the input value.
  3. Work out each term on its own.
  4. Add and subtract the terms to get the output.

Step 1

Step 1 - Set up the problem.

In this problem: The function rule is \(f(x) = 2x - 5\) and the input is \(x = -3\).

\(f(x) = 2x - 5,\; x = -3\)

Step 2

Step 2 - Replace every x in the function rule with the input value.

In this problem: Every \(x\) in the rule stands for the input, so each one becomes \(\left(-3\right)\), giving \(f\left(-3\right) = 2\left(-3\right) - 5\).

\(f\left(-3\right) = 2\left(-3\right) - 5\)

Step 3

Step 3 - Work out each term on its own.

In this problem: Working out each term on its own turns the expression into \(-6 - 5\).

\(f\left(-3\right) = -6 - 5\)

Step 4

Step 4 - Add and subtract the terms to get the output.

In this problem: Adding and subtracting \(-6 - 5\) leaves \(-11\).

\(f\left(-3\right) = -11\)

Final Answer

\(f\left(-3\right) = -11\)