Question

18 multiple choice 1 point which function does not have the set of all real numbers as its domain? \\( f(x) = |2x - 1| \\) \\( f(x) = \cos(x) + 1 \\) \\( f(x) = \frac{x + 1}{x + 3} \\) \\( f(x) = 5^x - 3 \\)

Explanation:

Step1: Analyze domain of absolute value

The function $f(x)=|2x-1|$ is defined for all real $x$, since absolute value accepts all real inputs.

Step2: Analyze domain of trigonometric function

The function $f(x)=\cos(x)+1$ is defined for all real $x$, as cosine is defined for all real numbers.

Step3: Analyze domain of rational function

For $f(x)=\frac{x+1}{x+3}$, the denominator cannot be 0. Set $x+3=0$, solve for $x$:
$x=-3$. So $x=-3$ is excluded from the domain.

Step4: Analyze domain of exponential function

The function $f(x)=5^x-3$ is defined for all real $x$, as exponential functions accept all real inputs.

Answer:

C. $f(x)=\frac{x+1}{x+3}$