Question 507·Hard·Algebra

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The linear function $f$ is defined by $f(x)=mx+b$ , where $m$ and $b$ are constants. The graph of $y=f(x)$ is perpendicular to the graph of $3x+2y=6$ . In addition, the composition $f(f(x))$ has an $x$ -intercept at $x=6$ . Which of the following is the value of $b$ ?

The cue is a perpendicular line paired with an intercept of a composition, which applies a function twice. Find the new slope, then set the output after both applications equal to zero. A Desmos regression finds the remaining constant. Flip the slope to make the lines perpendicular, but don’t flip the intercept after solving for it.

Hints

Hint 1
To read a line’s slope, get $y$ by itself. A perpendicular line has the negative reciprocal slope: flip the fraction and change its sign. What does that make $m$ ?
Hint 2
An $x$ -intercept is where a graph’s output is zero. Here the graph shows a function applied twice, so which output must equal zero: the first one or the second one?
Hint 3
A composition feeds one output into the function again. For $f(x)=mx+b$ , that gives $f(f(t))=m(mt+b)+b$ . Keep both copies of $b$ when you use the intercept condition.

Step-by-step

Find the slope, then solve the composition
Step 1
Read the given line’s slope
Put the given line in slope-intercept form, where the coefficient of $x$ is the slope. Subtract $3x$ :
$2y=-3x+6.$
Divide by $2$ :
$y=-\frac32x+3.$
So this line’s slope is $-\frac32$ .
Step 2
Find the perpendicular slope
Perpendicular lines have negative reciprocal slopes: flip $-\frac32$ and change its sign. So the slope of $f$ is $m=\frac23$ .
Step 3
Make the composite output zero
An $x$ -intercept is a point where the graph’s output is $0$ . The graph here is $y=f(f(x))$ , so its intercept at $x=6$ gives
$f(f(6))=0.$
The outer $f$ must return zero, not necessarily the inner $f(6)$ .
Step 4
Apply the function twice
First, $f(6)=\frac23(6)+b$ . A composition uses that whole output as the next input, so
$f(f(6))=\frac23\left(\frac23(6)+b\right)+b=0.$
The second $b$ comes from applying the rule again.
Step 5
Solve for the intercept in Desmos
Type $\frac23\left(\frac23\cdot6+b_1\right)+b_1\sim0$ . Here $b_1$ stands for $b$ , and $\sim$ tells Desmos to fit that unknown; don’t make a slider. Under PARAMETERS, Desmos shows $b_1=-1.6$ . Type $b_1$ on the next line and tap its fraction button to see $-\frac85$ . That is the $y$ -intercept $b$ of $f$ . Choice D.

Question 507·Hard·Algebra

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