Question

The domain of a reciprocal function is (−∞ ,3)∪(3, ∞)
. Which of the following functions g(x)
correctly denotes this shift?(1 point)
Responses

g(x)=1x+3
g left parenthesis x right parenthesis equals Start Fraction 1 over x plus 3 End Fraction

g(x)=1x−2
g left parenthesis x right parenthesis equals Start Fraction 1 over x minus 2 End Fraction

g(x)=1x−3
g left parenthesis x right parenthesis equals Start Fraction 1 over x minus 3 End Fraction

g(x)=1x+52

Answer 1

To determine which function \( g(x) \) corresponds to the given domain of the reciprocal function, we need to consider what values of \( x \) make the function undefined.

The domain of the reciprocal function is \( (-\infty, 3) \cup (3, \infty) \). This means that the function is undefined at \( x = 3 \).

The general form of a reciprocal function is \( g(x) = \frac{1}{x - a} \), where \( a \) is the value that makes the function undefined (the vertical asymptote).

Since we want the function to be undefined at \( x = 3 \), we can set \( a = 3 \). Therefore, the correct function is:

\[
g(x) = \frac{1}{x - 3}
\]

Now let's analyze the provided options:

1. \( g(x) = \frac{1}{x + 3} \) - undefined at \( x = -3 \)
2. \( g(x) = \frac{1}{x - 2} \) - undefined at \( x = 2 \)
3. **\( g(x) = \frac{1}{x - 3} \)** - undefined at \( x = 3 \) (correct option)
4. \( g(x) = \frac{1}{x + 52} \) - undefined at \( x = -52 \)

Thus, the correct response is:

\[
g(x) = \frac{1}{x - 3}
\]