Question 826·Medium·Math

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$g(x)=\dfrac{x^{2}-9}{x-3}$

When the graph of $y=g(x)$ is drawn in the $xy$ -plane, it appears as a straight line with a single hole (a point that is missing from the graph). What is the $y$ -coordinate of that hole?

Your answer

A removable discontinuity, or hole, occurs when a denominator factor cancels with a factor in the numerator. Find the excluded input from the original denominator, then evaluate the shorter expression there to get the hole’s height. Desmos can show that the graphs overlap, but a single missing point can be hard to see, so don’t rely on the picture to locate it.

Hints

Hint 1
A denominator is the bottom of a fraction, and division by $0$ is undefined. Set the original denominator equal to $0$ . Which input must the graph leave out?
Hint 2
A difference of squares follows $a^2-b^2=(a-b)(a+b)$ . Since $9=3^2$ , factor the numerator before canceling. You can cancel whole factors, not individual terms.
Hint 3
A hole is a point on the shorter line that the original graph leaves out. Evaluate that line at the excluded input to find the hole’s height, not the value of the original function there.

Step-by-step

Find the excluded input, then the line’s height
Step 1
Find the missing input
The original denominator cannot be $0$ . Set it equal to $0$ :
$x-3=0$
Add $3$ to both sides:
$x=3$
So $g$ has no point at this input, even if a shorter formula later allows it.
Step 2
Factor the numerator
The numerator is a difference of squares: $9=3^2$ , and $a^2-b^2=(a-b)(a+b)$ . So:
$x^2-9=(x-3)(x+3)$
The factor $x-3$ now matches the denominator.
Step 3
Cancel without restoring the missing point
Cancel the whole factor $x-3$ :
$\frac{(x-3)(x+3)}{x-3}=x+3,\qquad x\ne3$
The cancellation works only when $x-3$ is not $0$ . Canceling a factor does not cancel its restriction. So the graph follows $y=x+3$ but has a hole at $x=3$ .
Step 4
Compare the original and shorter graphs
Type $g(x)=\frac{x^2-9}{x-3}$ and $s(x)=x+3$ in Desmos. Their graphs overlap wherever $g$ is defined. The single missing point may be invisible on the graph, but $s$ gives the line’s height at that input.
Step 5
Evaluate the height at the missing input
Type $s(3)$ ; Desmos prints $6$ . This is the height of the hole, not $g(3)$ : the original fraction is undefined at $x=3$ . So the hole’s $y$ -coordinate is $6$ . Grid in 6.

Question 826·Medium·Math

Strategy

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