The linear function satisfies and , where and are distinct real numbers. Which of the following gives in terms of , , and ?

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SAT Linear Functions Question 87 | Free SAT Prep | Aniko

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Question 87·Hard·Linear Functions

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The linear function $g$ satisfies $g(a)=b$ and $g(b)=a$ , where $a$ and $b$ are distinct real numbers. Which of the following gives $g(x)$ in terms of $x$ , $a$ , and $b$ ?

For linear-function questions where you are given values like $g(a)$ and $g(b)$ , immediately interpret them as points on a line: $(a, g(a))$ and $(b, g(b))$ . Write the general form $g(x)=mx+k$ , substitute each given condition to create two equations, then use quick elimination or substitution to solve for $m$ and $k$ . Once you have the slope and intercept, rewrite $g(x)$ and match it directly to the answer choices; this avoids guessing and works efficiently even when the parameters are letters instead of numbers.

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Use the general form of a linear function

Think of $g(x)$ as a line with equation $g(x)=mx+k$ . How can you use the facts $g(a)=b$ and $g(b)=a$ with this form?

Turn function values into equations

Substitute $x=a$ and $x=b$ into $g(x)=mx+k$ to get two equations involving $m$ , $k$ , $a$ , and $b$ .

Eliminate one variable

You will get two equations in $m$ and $k$ . Try subtracting one equation from the other so that $k$ cancels and you can solve for $m$ .

Finish the function

Once you know $m$ , plug it back into one of your earlier equations to find $k$ , then write the full expression for $g(x)$ and compare it with the choices.

Desmos Guide

Set specific values for a and b

Pick two convenient distinct numbers for $a$ and $b$ (for example, $a=2$ and $b=5$ ), and in Desmos type a=2 and b=5 (or your chosen values) so they are defined.

Enter each answer choice as a function

For each option, type it into Desmos as a separate function, for example g1(x)=x+a+b, g2(x)=-x+a+b, g3(x)=(a+b)x-ab, and g4(x)=-x+a/b so you can test them one by one.

Test the conditions g(a)=b and g(b)=a

For each function, create expressions like g1(a), g1(b), g2(a), g2(b), etc. The correct function is the one for which the evaluated results satisfy both conditions: the value at $x=a$ equals your chosen $b$ , and the value at $x=b$ equals your chosen $a$ .

Step-by-step Explanation

Translate the information into points on the line

The function $g$ is linear and satisfies $g(a)=b$ and $g(b)=a$ .

That means the graph of $g$ is a straight line that passes through the two points:

$(a, b)$
$(b, a)$

Any linear function that passes through two points can be written in the slope-intercept form $g(x)=mx+k$ , where $m$ is the slope and $k$ is the $y$ -intercept.

Write equations using the general linear form

Assume $g(x)=mx+k$ .

Use the two conditions:

$g(a)=b$ becomes

ma + k = b

$g(b)=a$ becomes

mb + k = a

So we have a system of two equations in $m$ and $k$ :

ma + k = b \\[0.7em] mb + k = a

Solve for the slope m

Subtract the second equation from the first to eliminate $k$ :

(ma + k) - (mb + k) = b - a \\[0.7em] ma - mb = b - a \\[0.7em] m(a - b) = b - a

Because $a$ and $b$ are distinct, $a-b \neq 0$ , so we can divide both sides by $(a-b)$ :

m = \frac{b - a}{a - b}

Notice that $b-a = -(a-b)$ , so

m = \frac{-(a - b)}{a - b} = -1

Find the intercept k and write g(x)

Now plug $m=-1$ into one of the equations, for example $ma + k = b$ :

(-1)a + k = b \\[0.7em] -a + k = b

Add $a$ to both sides:

k = a + b

So the linear function is

g(x) = -x + (a + b)

which matches the choice $g(x) = -x + a + b$ .

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Step-by-step Explanation

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Step-by-step Explanation