Question 25·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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Let $k$ be a positive constant. The graph of the equation below has no $x$ -intercepts.

y=\frac{3^{2x}-(k+9)3^x+9k}{3^x-k}

Which choice could be the value of $k$ ?

A rational function containing both $a^{2x}$ and $a^x$ may hide a factor that cancels with the denominator. Graph a trial value in Desmos to spot a possible $x$ -intercept, then factor the numerator and check which inputs the original denominator forbids. The missing-point trap is assuming that canceling a factor restores an excluded point.

Hints

Hint 1
An $x$ -intercept is a point where $y=0$ . Graph one trial value of $k$ to spot a possible intercept, but check the denominator before deciding whether that point belongs to the original graph.
Hint 2
Since $3^{2x}=(3^x)^2$ , the numerator is a quadratic in $3^x$ . Try factoring it as you would a quadratic in a single letter. Does one factor match the denominator?
Hint 3
The domain is the set of allowed inputs. Canceling a factor doesn't restore an input that makes the original denominator $0$ . Where must the possible intercept fall for the graph to have none?

Step-by-step

Find the intercept, then check the missing point
Step 1
Spot a possible intercept
An $x$ -intercept has $y=0$ . Test the choice $k=3$ in Desmos: type $k=3$ , then type the given equation. Click the crossing at $(2,0)$ . This gives you a candidate input to investigate, not a conclusion about every value of $k$ .
Step 2
Keep the original restriction
The original denominator is $3^x-k$ . It equals $0$ when $3^x=k$ , so that input is excluded from the graph, even if the same factor later cancels.
Step 3
Factor the repeated power
Because $3^{2x}=(3^x)^2$ , treat $3^x$ as one quantity. Factor the numerator:
$3^{2x}-(k+9)3^x+9k=(3^x-k)(3^x-9).$
The middle terms combine to $-(k+9)3^x$ , and one factor matches the denominator.
Step 4
Cancel without restoring the excluded input
Cancel the matching factor only where it isn't $0$ :
$y=\frac{(3^x-k)(3^x-9)}{3^x-k}=3^x-9,\qquad 3^x\ne k.$
Canceling a factor doesn't bring back its excluded input.
Step 5
Find the only possible intercept
Set the simplified output to $0$ to find where an intercept could be:
$3^x-9=0.$
Add $9$ :
$3^x=9.$
Since $9=3^2$ , equal powers of base $3$ have equal exponents. So only $x=2$ could give an intercept.
Step 6
Exclude that one point
At $x=2$ , the original denominator is $3^2-k=9-k$ . Make it $0$ by taking $k=9$ . Then $(2,0)$ is missing from the graph, and there are no other possible $x$ -intercepts. The value of $k$ is $9$ . Choice C.

Question 25·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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