Question

18 describe the domain and range for the function ( y = \frac{1}{2}(4)^{x + 2}-5 ). a domain: ( x > -2 ) range: ( y > 4 ) b domain: all real numbers range: ( y < 5 ) c domain: all real numbers range: ( y > -5 ) d domain: ( x < -2 ) range: all real numbers

Explanation:

Step1: Identify function type

This is an exponential function $y=\frac{1}{2}(4)^{x+2}-5$. Exponential functions of the form $a(b^{cx+d})+k$ have no restrictions on $x$ (no division by zero, no negative under root for real numbers).

Step2: Find the domain

For all real values of $x$, $4^{x+2}$ is defined. Thus, domain is all real numbers.

Step3: Analyze the exponential term

The term $4^{x+2}$ is an exponential function with base $4>1$, so $4^{x+2} > 0$ for all real $x$.

Step4: Transform to find range

Multiply by $\frac{1}{2}$: $\frac{1}{2}(4)^{x+2} > 0$. Subtract 5: $\frac{1}{2}(4)^{x+2}-5 > -5$, so $y > -5$.

Answer:

C. Domain: all real numbers
Range: y > -5