Evaluating a Function

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Idea

Evaluating a function means substituting the entire input expression wherever the variable appears. Parentheses preserve the structure, especially when the input is negative or itself a sum.

Visual guide

VISUAL GUIDEFollow the input through the rule

f(x) = 2x + 1

For f(x) = 2x + 1, substitution sends −1 to −1, 0 to 1, and 2 to 5. The function value is the output at the arrow’s destination.

Method

For $f(x)=x^2-3x$ , input $u$ gives $f(u)=u^2-3u$ . If $u$ is an expression, expand only after substitution. Check domain restrictions first.

Worked example

Compute $f(a+h)$ :

f(a+h)=(a+h)^2-3(a+h)=a^2+2ah+h^2-3a-3h.

Subtracting $f(a)$ leaves $2ah+h^2-3h$ . If $h\ne0$ , dividing by $h$ gives $2a+h-3$ , the difference quotient used in differentiation.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If f(x)=2x²−1, what is f(−3)?

−19

Hint 1 · Find a starting point

Substitute the whole input, including its sign.

Hint 2 · Take the next step

(−3)² = 9.

Show the reasoning

Answer: 17

2(−3)²−1 = 18−1 = 17.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

$f(a+h)$ is usually not $f(a)+f(h)$ . A nonlinear function does not distribute over addition.

Check your understanding

For $g(x)=1/(x-2)$ , find $g(5)$ and state why $g(2)$ fails.

Show answer

$g(5)=1/3$ ; at $2$ the denominator is zero.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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