Question 327·Hard·Algebra

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The linear function $g$ passes through the points $(2,-3)$ and $(8,15)$ .

A new function is defined by

r(x)=g(3x-4)+2.

Which equation defines $r$ ?

A line given by two points and used inside a new rule signals function composition. Fit the original line with a Desmos regression, then evaluate the new function at $0$ and $1$ . Since the result is still a line, its outputs at $0$ and $1$ give its intercept and slope. Don't reuse the old slope: changing the input can change it.

Hints

Hint 1
A linear function can be written $g(x)=mx+b$ , where $m$ is the output change for each $1$ added to the input and $b$ is the output at $0$ . Two points determine both numbers; can you fit them?
Hint 2
The input of $g$ is everything inside its parentheses. In $g(3x-4)$ , the entire expression $3x-4$ replaces the input in $g$ 's rule. The outside $+2$ is added afterward. What rule does that make?
Hint 3
For a linear function, the output at $0$ is its intercept, and the output change from $0$ to $1$ is its slope. Evaluate the new function, not the original, at those two inputs.

Step-by-step

Fit the line, then use two new outputs
Step 1
Find the rule for $g$
Enter the two given points in a Desmos table. A linear function has the form $g(x)=mx+b$ , where $m$ is its slope, or output change per input unit, and $b$ is its output at $0$ . Type $y_1\sim mx_1+b$ to fit that form. Under PARAMETERS, Desmos shows $m=3$ and $b=-9$ , so $g(x)=3x-9$ .
Step 2
Put the whole new input into $g$
Replace $g$ 's input with the whole expression $3x-4$ , then add $2$ to the output: $r(x)=3(3x-4)-9+2$ . To keep that composition in Desmos, type $g(x)=mx+b$ and $r(x)=g(3x-4)+2$ . The $-4$ is inside $g$ ; the $+2$ is outside.
Step 3
Find $r$ 's intercept
The $y$ -intercept is the output when the input is $0$ . Type $r(0)$ ; Desmos prints $-19$ , so $r$ passes through $(0,-19)$ . Don't use $b=-9$ here: that's $g$ 's intercept. In $r$ , an input of $0$ goes into $g$ as $-4$ .
Step 4
Find one more output of $r$
Type $r(1)$ ; Desmos prints $-10$ , giving the point $(1,-10)$ . Because both $g$ and its new input $3x-4$ are linear, $r$ is also a line. These two outputs, at inputs $0$ and $1$ , will give its slope.
Step 5
Use the slope and intercept to write $r$
Type $r(1)-r(0)$ ; Desmos prints $9$ . The inputs differ by one unit, so this output change is $r$ 's slope. Its intercept is $-19$ , so the equation defining the new function is $r(x)=9x-19$ . Choice A.

Question 327·Hard·Algebra

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