To determine the range of the function [tex]\( f(x) = \frac{3}{4} |x| - 37 \)[/tex], let's analyze the function step-by-step.
1. Absolute Value Component: The function [tex]\( f(x) \)[/tex] includes an absolute value term [tex]\( |x| \)[/tex]. The absolute value [tex]\( |x| \)[/tex] is always non-negative ([tex]\( |x| \geq 0 \)[/tex]) for all real numbers [tex]\( x \)[/tex].
2. Multiplication by Constant: Next, consider the term [tex]\( \frac{3}{4} |x| \)[/tex]. Since [tex]\( |x| \geq 0 \)[/tex], multiplying by [tex]\( \frac{3}{4} \)[/tex] (a positive constant) does not change the non-negative property. Hence, [tex]\( \frac{3}{4} |x| \geq 0 \)[/tex].
3. Shifting the Function: The function then subtracts 37 from [tex]\( \frac{3}{4} |x| \)[/tex]. Therefore, [tex]\( f(x) = \frac{3}{4} |x| - 37 \)[/tex].
- When [tex]\( |x| = 0 \)[/tex], [tex]\( f(x) = \frac{3}{4} \cdot 0 - 37 = -37 \)[/tex].
- As [tex]\( |x| \)[/tex] increases (becomes larger), [tex]\( \frac{3}{4} |x| \)[/tex] also increases. Therefore, [tex]\( f(x) = \frac{3}{4} |x| - 37 \)[/tex] also increases.
4. Lower Bound: The minimum value of [tex]\( \frac{3}{4} |x| \)[/tex] is 0 (when [tex]\( |x| = 0 \)[/tex]). This makes the minimum value of [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) \big|_{|x|=0} = \frac{3}{4} \cdot 0 - 37 = -37.
\][/tex]
5. Upper Bound: There is no upper bound for [tex]\( f(x) \)[/tex] because as [tex]\( |x| \)[/tex] becomes larger, [tex]\( \frac{3}{4} |x| \)[/tex] can grow without limit. Hence, [tex]\( f(x) \)[/tex] can become arbitrarily large.
Range Conclusion: Considering these points, the function [tex]\( f(x) = \frac{3}{4} |x| - 37 \)[/tex] takes values starting from the minimum value which is -37 and can go up to any larger value. Thus, the range of the function is:
[tex]\[
\boxed{ \text{all real numbers greater than or equal to } -37 }
\][/tex]
