Question

Let $f\left(x\right)=\frac{3}{x+7}+4$ be a function. Then

• A. domain of $f$ is $R-\{-7\}$
• B. domain of $f$ is $R$
• C. range of $f$ is $R-\left\{4\right\}$
• D. range of $f$ is $R$

Answer 1

Answer: (A), (C)

The correct options are
A domain of $f$ is $R-\{-7\}$
C range of $f$ is $R-\left\{4\right\}$

$f\left(x\right)=\frac{3}{x+7}+4$
Clearly, $f$ is not defined at $x=-7$
Hence, domain of $f$ is $R-\{-7\}$
Let $y=\frac{3}{x+7}+4$
$\Rightarrow y-4=\frac{3}{x+7}$
$\Rightarrow x+7=\frac{3}{y-4}$
$\Rightarrow x=\frac{3}{y-4}-7$
Clearly, $x$ is defined for all $y\in R-\left\{4\right\}$
Hence, range of $f$ is $R-\left\{4\right\}$