What is the range of the function f(x)=3/4|x|-3

To find the range of the function \( f(x) = \frac{3}{4} |x| - 3 \), we can analyze the behavior of the function for different values of \( x \). The absolute value function \( |x| \) ensures that the expression inside the absolute value is always non-negative.

Let's break it down:

1. When \( x \geq 0 \):

  \[ f(x) = \frac{3}{4}x - 3 \]

2. When \( x < 0 \):

  \[ f(x) = \frac{3}{4}(-x) - 3 = -\frac{3}{4}x - 3 \]

Now, consider the behavior of each part separately:

- For \( x \geq 0 \), the function \( f(x) = \frac{3}{4}x - 3 \) is a linear function with a positive slope. As \( x \) increases, \( f(x) \) increases without bound.

- For \( x < 0 \), the function \( f(x) = -\frac{3}{4}x - 3 \) is also a linear function, but with a negative slope. As \( x \) becomes more negative, \( f(x) \) decreases without bound.

Therefore, the range of the function is all real numbers. In interval notation, this can be expressed as \( (-\infty, \infty) \).