Let's determine the range of the function [tex]\( f(x) = \frac{3}{4} |x| - 3 \)[/tex].
1. Understand the Function: The function involves an absolute value. Remember that the absolute value function [tex]\(|x|\)[/tex] always produces a non-negative result.
2. Analyze the Expression [tex]\(\frac{3}{4} |x|\)[/tex]: Since [tex]\(|x|\)[/tex] is non-negative, [tex]\(\frac{3}{4} |x|\)[/tex] is also non-negative. The smallest value [tex]\(\frac{3}{4} |x|\)[/tex] can take is 0, which occurs when [tex]\(x = 0\)[/tex].
3. Add the Constant Term (-3): We subtract 3 from [tex]\(\frac{3}{4} |x|\)[/tex]. Since the minimum value of [tex]\(\frac{3}{4} |x|\)[/tex] is 0, the minimum value of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(0) = \frac{3}{4} \cdot 0 - 3 = -3 \][/tex]
4. Determine the Range: As [tex]\(|x|\)[/tex] grows larger, [tex]\(\frac{3}{4} |x|\)[/tex] increases indefinitely. Thus, [tex]\( f(x) \)[/tex] will also increase indefinitely since we are subtracting a constant (-3) from a growing positive term. Therefore, the function can take on all values greater than or equal to [tex]\(-3\)[/tex].
So, the range of the function [tex]\( f(x) = \frac{3}{4} |x| - 3 \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers greater than or equal to } -3} \][/tex]
