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------------------------------------------------ The function [tex]f(x)=-\frac{1}{4} x-3[/tex] is defined for values of [tex]x[/tex] on the interval [tex]-4 < x < 16[/tex].

What are the domain and range of [tex]f(x)[/tex]?

A. Domain: all real numbers; Range: all real numbers

B. Domain: [tex]-4 < x < 16[/tex]; Range: [tex]4 < f(x) < 7[/tex]

C. Domain: [tex]-4 < x < 16[/tex]; Range: [tex]-7 < f(x) < -2[/tex]

D. Domain: [tex]0 < x < 4[/tex]; Range: [tex]-4 < f(x) < -7[/tex]

Let's break down the problem to find the domain and range of the function [tex]$ f(x) = -\frac{1}{4} x - 3 $[/tex] given the interval [tex]$ -4 < x < 16 $[/tex].

Domain:
The domain specifies the set of input values (x-values) for which the function is defined. According to the problem, the domain is given as [tex]$ -4 < x < 16 $[/tex]. So we clearly omit options A and D.

Range:
To find the range, we need to determine the set of output values (y-values) that the function [tex]$ f(x) $[/tex] can take when [tex]$ x $[/tex] is within the given interval.

1. First, calculate the value of [tex]$ f(x) $[/tex] at the endpoints of the interval:
- At [tex]$ x = -4 $[/tex]:
[tex]\[
f(-4) = -\frac{1}{4}(-4) - 3 = 1 - 3 = -2
\][/tex]

- At [tex]$ x = 16 $[/tex]:
[tex]\[
f(16) = -\frac{1}{4}(16) - 3 = -4 - 3 = -7
\][/tex]

2. Observing these values shows us that [tex]$ f(-4) = -2 $[/tex] is the highest point and [tex]$ f(16) = -7 $[/tex] is the lowest point.

3. Therefore, the range of the function [tex]$ f(x) $[/tex] as [tex]$ x $[/tex] goes from -4 to 16 is [tex]$ -2 > f(x) > -7 $[/tex], which translates to the interval [tex]$ -7 < f(x) < -2 $[/tex].

Given these observations, the correct choice is:

C. domain: [tex]$-4 < x < 16$[/tex]; range: [tex]$-7 < f(x) < -2$[/tex]