To determine the range of the function \( f(x) = \left(\frac{3}{4}\right)^x - 4 \), let's analyze the behavior of the function step-by-step.
1. Understand the function structure:
The given function is of the form \( f(x) = a^x - 4 \), where \( a \) is the base of the exponential function and in this case \( a = \frac{3}{4} \).
2. Behavior of the exponential term:
The exponential term is \( \left(\frac{3}{4}\right)^x \). Since \( \frac{3}{4} \) is less than 1:
- As \( x \) approaches positive infinity (\( x \to \infty \)), \( \left(\frac{3}{4}\right)^x \) approaches 0.
- As \( x \) approaches negative infinity (\( x \to -\infty \)), \( \left(\frac{3}{4}\right)^x \) grows very large because raising a fraction to a negative power results in a larger number.
3. Analyze the limits:
- When \( x \to \infty \):
\( f(x) = \left(\frac{3}{4}\right)^x - 4 \) approaches \( 0 - 4 = -4 \).
- When \( x \to -\infty \):
\( f(x) = \left(\frac{3}{4}\right)^x - 4 \) can be extremely large (since \( \left(\frac{3}{4}\right)^{-x} \) becomes very large as \( x \to -\infty \)) minus 4, but this value will still be large and positive.
4. Determine the range:
- From the analysis,
- \( f(x) \) approaches but never actually reaches -4, meaning \( f(x) \) is always greater than -4 as \( x \to \infty \).
- As \( x \to -\infty \), \( f(x) \) increases without bound.
Therefore, the output values of \( f(x) \) are all values greater than -4. Thus, the range of \( f(x) = \left(\frac{3}{4}\right)^x - 4 \) is:
[tex]\[ \{ y \mid y > -4 \} \][/tex]
So, the correct answer is:
[tex]\[ \{ y \mid y > -4 \} \][/tex]
