Question

question 6 of 10
select all the correct locations on the image.
what are the domain and the range of this function?
$f(x) = \frac{3}{x + 5} - 7$
| ${y|y \in \mathbb{r}, y \
eq -7}$ | ${y|-7 < y < \infty}$ | ${y|y \in \mathbb{r}, y \
eq -5}$ |

${x

-5 < x < \infty}$

${x

x \in \mathbb{r}, x \

eq -5}$ | ${x|x \in \mathbb{r}, x \
eq -7}$ |

Explanation:

Step1: Find the domain of \( f(x)=\frac{3}{x + 5}-7 \)

The function has a rational part \( \frac{3}{x + 5} \). The denominator cannot be zero. So, we set \( x+5
eq0 \), which gives \( x
eq - 5 \). The domain is all real numbers except \( x=-5 \), so the domain is \( \{x|x\in\mathbb{R},x
eq - 5\} \).

Step2: Find the range of \( f(x)=\frac{3}{x + 5}-7 \)

Let \( y = \frac{3}{x + 5}-7 \). We can rewrite it as \( y + 7=\frac{3}{x + 5} \), then \( x + 5=\frac{3}{y + 7} \). For \( x \) to be defined, the denominator \( y + 7
eq0 \), so \( y
eq - 7 \). The range is all real numbers except \( y = - 7 \), so the range is \( \{y|y\in\mathbb{R},y
eq - 7\} \).

Answer:

Domain: \( \{x|x\in\mathbb{R},x
eq - 5\} \)
Range: \( \{y|y\in\mathbb{R},y
eq - 7\} \)