Linear Functions Pattern - Direct Calculation

Plugging in a value to a linear function to find the output or input

These questions give you a linear function — either in function notation like $f(x) = 3x + 5$ or as an equation like $C = 8m + 50$ — and ask you to substitute a value in and compute. Sometimes you plug in the input to find the output ("find $f(4)$"). Other times you're given the output and must solve for the input ("if $f(k) = 21$, find $k$"). Either way, the math is straightforward substitution and arithmetic.

 

Finding the Output (Forward Substitution)

The most basic version: replace the variable with the given number and simplify.

The function $p$ is defined by $p(y) = 50 - 4y$. What is the value of $p(8)$?

Substitute $8$ for $y$:

$$p(8) = 50 - 4(8) = 50 - 32 = \boldsymbol{18}$$

Watch the order of operations — multiply before you add or subtract. A common wrong answer here would be $50 - 4 = 46$ (forgetting the input) or $50 - 48 = 2$ (misreading $4y$ as $48$).

When the input is negative, be careful with signs:

The function $T$ is defined by $T(c) = 2c + 77$. What is the value of $T(c)$ when $c = -5$?

Substitute $-5$ for $c$:

$$T(-5) = 2(-5) + 77 = -10 + 77 = \boldsymbol{67}$$

A frequent mistake is stopping at $2(-5) = -10$ without adding $77$, or doing $(2 + 77)(-5) = -395$ by adding before multiplying.

 

Context Problems

The SAT often wraps the same substitution in a real-world scenario. The function models something (cost, distance, temperature), and you plug in a given value.

The volume $V$, in gallons, of water in a tank $t$ minutes after a valve is opened is given by $V = 200 + 15t$. What is the volume of water in the tank 10 minutes after the valve is opened?

Substitute $t = 10$:

$$V = 200 + 15(10) = 200 + 150 = \boldsymbol{350}$$

The wrong answers on these problems test whether you confused the slope with the intercept. Here, $200$ would mean you forgot to multiply, and $215$ would mean you only added one unit of time.

 

Finding the Input (Reverse Substitution)

Instead of giving you the input, the question gives you the output and asks you to solve for the input. Set the function equal to the given value and solve the resulting equation.

A function $k$ is defined as $k(n) = \dfrac{1}{2}n + 11$. Find the value of $n$ such that $k(n) = 19$.

Set the function equal to $19$:

$$\frac{1}{2}n + 11 = 19$$

Subtract $11$: $\dfrac{1}{2}n = 8$. Multiply both sides by $2$: $n = \boldsymbol{16}$.

A trap answer is $n = 4$, from dividing $8$ by $2$ instead of multiplying. Another is $n = 20.5$, from plugging $19$ in as the input instead of the output.

Here's one with a fraction in the function definition:

The function $h$ is defined by $h(p) = \dfrac{20 + p}{2}$. Given that $h(c) = 18$, what is the value of $c$?

Set the function equal to $18$:

$$\frac{20 + c}{2} = 18$$

Multiply both sides by $2$: $20 + c = 36$. Subtract $20$: $c = \boldsymbol{16}$.

 

Finding Intercepts

When the question asks for the $y$-intercept of $y = f(x)$, you're really just being asked to compute $f(0)$. Plug in $x = 0$ and evaluate.

The function $g$ is defined by $g(x) = 5x + 8$. What is the y-intercept of the graph of $y = g(x)$?

The $y$-intercept occurs when $x = 0$:

$$g(0) = 5(0) + 8 = 8$$

The $y$-intercept is $\boldsymbol{(0, 8)}$.

For a function in slope-intercept form $y = mx + b$, the $y$-intercept is always the constant $b$. You don't even need to substitute — just read it off. But the SAT offers the slope as a wrong answer, so make sure you grab the right number.

The $x$-intercept works the opposite way — set $f(x) = 0$ and solve for $x$:

The function $g$ is defined by $g(x) = 4x + 52$. What is the x-intercept of the graph of $y = g(x)$?

Set the function equal to $0$:

$$4x + 52 = 0$$

Subtract $52$: $4x = -52$. Divide by $4$: $x = -13$. The $x$-intercept is $\boldsymbol{(-13, 0)}$.

 

Constant Functions

A sneaky medium-difficulty variant: the function is defined as a constant, like $g(x) = -12$. There's no $x$ in the expression, so the output is $-12$ no matter what the input is. The SAT shows you four tables and asks which one matches.

The linear function $g$ is defined by $g(x) = -12$. Which table shows three values of $x$ and their corresponding values of $g(x)$?

Since $g(x) = -12$ for all $x$, every row of the table must have $-12$ in the output column. The correct table is:

$x$	$g(x)$
$0$	$-12$
$1$	$-12$
$2$	$-12$

The main trap is confusing $g(x) = -12$ with $g(x) = -12x$. The first is constant (horizontal line); the second has slope $-12$.

 

Approximation with Decimals

Medium-level context problems sometimes use decimal coefficients and ask for the "closest" value. The substitution is the same — just be careful with the arithmetic.

The value of a car is modeled by $V = -2.4x + 35$, where $V$ is in thousands of dollars and $x$ is years since purchase. If the car was purchased 4 years ago, which is closest to the predicted value?

Substitute $x = 4$:

$$V = -2.4(4) + 35 = -9.6 + 35 = 25.4$$

The closest value is $\boldsymbol{25}$ thousand dollars.

The wrong answer $45$ comes from ignoring the negative sign ($2.4 \times 4 + 35 = 44.6$). Always pay attention to whether the coefficient is positive or negative.

 

Hard Problems: Complex Formulas

On hard questions, the function looks more intimidating — fractions with grouped terms, decimals, and more steps to unwind. But the process is identical: substitute and solve.

The rate of a chemical reaction, in moles per liter per second, is modeled by $R = \dfrac{0.05(C - 25)}{2} + 0.15$, where $C$ is the concentration of a catalyst. If the observed rate is $1.4$, what is $C$?

Set $R = 1.4$:

$$\frac{0.05(C - 25)}{2} + 0.15 = 1.4$$

Subtract $0.15$: $\dfrac{0.05(C - 25)}{2} = 1.25$

Multiply by $2$: $0.05(C - 25) = 2.5$

Divide by $0.05$: $C - 25 = 50$

Add $25$: $C = \boldsymbol{75}$

The algebra is longer, but each step is just undoing one operation. Work from the outside in: first undo the addition, then the division, then the multiplication, then the subtraction.

 

The Substitution Checklist

1. Read the question. Are you finding the output (given input) or the input (given output)?
2. Substitute. Replace the variable with the given value. Use parentheses around negative numbers.
3. Follow order of operations. Multiply/divide before you add/subtract.
4. If solving for input: undo operations one at a time, working from the outside in.
5. Check the answer format. Intercept questions want a coordinate pair like $(0, 8)$, not just the number $8$.

 

Watch Out For

• Order of operations errors. In $f(x) = -6x + 15$, the SAT offers $39$ as a wrong answer — that's what you get if you compute $(-6 + 15) \times 3$ instead of $-6(3) + 15$. Always multiply the coefficient by the input first.
• Sign errors with negatives. When plugging in a negative number, wrap it in parentheses: $2(-5)$, not $2 \cdot -5$. And remember that subtracting a negative is adding: $15 - (-5) = 20$.
• Confusing input and output. If $f(k) = 21$, the question asks for $k$, not for $f(21)$. Read carefully to see which direction you're going.
• Mixing up slope and intercept. In $g(x) = 5x + 8$, the $y$-intercept is $8$, not $5$. The SAT always offers the slope as a distractor when asking for the intercept.