To determine the range of the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex], let's analyze its behavior as [tex]\( x \)[/tex] varies.
1. Understand the function [tex]\( \left(\frac{3}{4}\right)^x \)[/tex]:
- [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] is an exponential function with a base less than 1.
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]), [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] approaches 0.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] tends towards positive infinity because for very large negative [tex]\( x \)[/tex], [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] gets larger and larger.
2. Analyze the function [tex]\( \left(\frac{3}{4}\right)^x - 4 \)[/tex]:
- When [tex]\( x \to +\infty \)[/tex], [tex]\( \left(\frac{3}{4}\right)^x \to 0 \)[/tex]. Therefore,
[tex]\[
\left(\frac{3}{4}\right)^x - 4 \to -4.
\][/tex]
- When [tex]\( x \to -\infty \)[/tex], [tex]\( \left(\frac{3}{4}\right)^x \to +\infty \)[/tex]. Thus,
[tex]\[
\left(\frac{3}{4}\right)^x - 4 \to +\infty - 4 = +\infty.
\][/tex]
3. Determine the range:
- Since [tex]\( \left(\frac{3}{4}\right)^x - 4 \)[/tex] approaches [tex]\(-4\)[/tex] as [tex]\( x \to +\infty \)[/tex] and increases without bound as [tex]\( x \to -\infty \)[/tex], the values this function can take are all real numbers greater than [tex]\(-4\)[/tex].
Therefore, the range of the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex] is:
[tex]\[
\{ y \mid y > -4 \}
\][/tex]
So the correct answer is:
[tex]\[
\{ y \mid y > -4 \}
\][/tex]
