Evaluation of Functions

Evaluating Nonlinear Functions

This pattern gives you a function definition — polynomial, exponential, radical, or rational — and asks you to compute a specific output. Plug in the given input, follow order of operations, and simplify. These are straightforward computation problems, but careless arithmetic is the #1 source of errors.

Worked Examples

Example 1. $h(t) = t^2 - 10$. What is $h(9)$?

A) $8$
B) $71$
C) $91$
D) $-1$

$h(9) = 9^2 - 10 = 81 - 10 = 71$
Gotcha: Option A computes $9 \times 2 - 10 = 8$ (multiplies instead of squaring). Option C forgets to subtract: $9^2 = 81$, but then adds $10$. Squaring means multiplying the number by itself, not by the exponent.
The answer is B.

Example 2. $g(x) = 2x^3$. What is $g(5)$?

A) $30$
B) $125$
C) $1000$
D) $250$

$g(5) = 2 \times 5^3 = 2 \times 125 = 250$
Gotcha: Option A computes $2 \times 5 \times 3 = 30$. Option B computes $5^3 = 125$ but forgets to multiply by $2$. Option C computes $2^3 \times 5^3 = 1000$ (cubes both the coefficient and the input). Exponents only apply to the base they're attached to.
The answer is D.

Example 3. $p(y) = \dfrac{100}{y - 5}$. What is $p(25)$?

A) $-1$
B) $4$
C) $20$
D) $5$

$p(25) = \dfrac{100}{25 - 5} = \dfrac{100}{20} = 5$
Gotcha: Option C divides $100$ by $5$ instead of by $25 - 5 = 20$. Always compute the denominator first, then divide. Option B divides $25$ by something instead of $100$.
The answer is D.

Example 4. $k(n) = 3 + 2^n$. What is $k(4)$?

A) $19$
B) $11$
C) $16$
D) $625$

$k(4) = 3 + 2^4 = 3 + 16 = 19$
Gotcha: Option B computes $3 + 2 \times 4 = 11$ (treats the exponent as multiplication). Option C gives $2^4 = 16$ but forgets to add $3$. Option D computes $(3 + 2)^4 = 625$ (adds before exponentiating — wrong order of operations).
The answer is A.

Example 5. $f(x) = \sqrt[3]{5x + 7}$. What is $f(4)$?

$f(4) = \sqrt[3]{5(4) + 7} = \sqrt[3]{27} = 3$
Evaluate inside the radical first: $5(4) + 7 = 27$. Then take the cube root: $\sqrt[3]{27} = 3$.

Example 6. A function is defined by $f(x) = -2(x - 3)^2 + 8$. What is $f(0)$?

$f(0) = -2(0 - 3)^2 + 8 = -2(9) + 8 = -18 + 8 = -10$
This gives the $y$-intercept. The $y$-intercept is always $f(0)$.

What to Do on Test Day

• Order of operations matters. Exponents before multiplication. Parentheses first. $2 \times 5^3 = 2 \times 125 = 250$, not $10^3 = 1000$.
• Squaring vs. multiplying by 2: $x^2$ means $x \times x$. It does not mean $2x$. This is the most common error on easy questions.
• Evaluate the denominator first in rational functions. $\dfrac{100}{25-5} = \dfrac{100}{20}$, not $\dfrac{100}{5}$.
• Negative inputs: $(-3)^2 = 9$ (positive), but $-3^2 = -9$ (negative). Parentheses matter.
• $y$-intercept = $f(0)$. Plug in $x = 0$ to find the $y$-intercept. For polynomial $ax^2 + bx + c$, the $y$-intercept is just $c$.
• Check your arithmetic. These problems are easy conceptually but punish careless mistakes. Double-check your computation before selecting an answer.