Start with $f(x)=mx+b$. Replace the $x$ inside $f$ with $mx+b$ to get a formula for $f(f(x))$ in terms of $m$, $b$, and $x$.

If the graph of $y=f(f(x))$ has an x-intercept at $x=6$, what must the value of $f(f(6))$ be? Use that to create an equation that involves $m$, $b$, and the number $6$.

Once you know $m$, plug it into your equation from the x-intercept condition and carefully solve the resulting linear equation for $b$.

Type y = (-3/2)x + 3 into Desmos to see the given line. Its slope is $-3/2$, so a perpendicular line must have slope $2/3$ (negative reciprocal). Use this value for $m$ in the next steps.

In Desmos, enter m = 2/3, then b as a slider. Define f(x) = m*x + b, and then define g(x) = f(f(x)).

Create a point at $(6, g(6))$ by entering (6, g(6)). Adjust the slider for b until this point lies exactly on the x-axis (its y-coordinate is 0). The corresponding value of b is the one you want; match it to the correct choice.

Type the equation 6*(2/3)^2 + b*((2/3) + 1) = 0 into Desmos, then wrap it in solve(..., b) to have Desmos solve for b. The solution it gives for b is the correct option.

First, rewrite $3x+2y=6$ in slope-intercept form.

So the slope of this line is $-\tfrac{3}{2}$. For a perpendicular line, the slopes are negative reciprocals, so $m$ must satisfy

This gives $m = \tfrac{2}{3}$, so $f(x) = \tfrac{2}{3}x + b$. (We still need $b$.)

Start with $f(x) = mx + b$ in general. Then

Simplify this:

Now we know $m = \tfrac{2}{3}$, so we will substitute this value for $m$ next.

An x-intercept at $x=6$ for the graph of $y = f(f(x))$ means that when $x=6$, the output is $0$:

Using $f(f(x)) = m^2 x + mb + b$:

Now substitute $m = \tfrac{2}{3}$:

First compute $\left(\tfrac{2}{3}\right)^2 = \tfrac{4}{9}$, then multiply by $6$:

So the equation becomes

Combine the $b$ terms:

so

Multiply both sides by $3$ to clear denominators:

Solve for $b$:

So the correct value of $b$ is $-\tfrac{8}{5}$.