Answer :
To determine the range of the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex], let's break it down step-by-step.
1. Exponential Function Behavior:
- The function [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] is an exponential function where the base is [tex]\(\frac{3}{4}\)[/tex], which is a fraction between 0 and 1.
- For exponential functions of the form [tex]\(a^x\)[/tex] where [tex]\(0 < a < 1\)[/tex], as [tex]\(x\)[/tex] increases, [tex]\(a^x\)[/tex] decreases. For example:
- When [tex]\( x = 0 \)[/tex], [tex]\( \left(\frac{3}{4}\right)^0 = 1 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x \to 0 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x\)[/tex] grows without bound since [tex]\(\left(\frac{3}{4}\right)^{-x} = \left(\frac{4}{3}\right)^x\)[/tex].
2. Shifting the Function Downward:
- The given function is shifted downward by 4 units. So, [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex].
- This shift affects the range of the function. We subtract 4 from each value [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] can take.
3. Finding the Range:
- The exponential function [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] can vary from 0 (exclusive) to 1 (inclusive):
- When [tex]\( x \to \infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x \to 0\)[/tex], so [tex]\( \left(\frac{3}{4}\right)^x - 4 \to -4 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x = 1\)[/tex], so [tex]\( f(x) = 1 - 4 = -3\)[/tex].
- Therefore, the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex] gets arbitrarily close to [tex]\(-4\)[/tex] but never actually reaches it, and it can grow without bound as [tex]\( x \)[/tex] moves towards [tex]\(-\infty \)[/tex].
4. Conclusion:
- The range of the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex] is all real numbers greater than [tex]\(-4\)[/tex]. This can be written in the set notation as [tex]\(\{y \mid y > -4\}\)[/tex].
Given these observations, the correct range of the function is:
[tex]\[ \{y \mid y > -4\} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\{y \mid y > -4\}} \][/tex]
1. Exponential Function Behavior:
- The function [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] is an exponential function where the base is [tex]\(\frac{3}{4}\)[/tex], which is a fraction between 0 and 1.
- For exponential functions of the form [tex]\(a^x\)[/tex] where [tex]\(0 < a < 1\)[/tex], as [tex]\(x\)[/tex] increases, [tex]\(a^x\)[/tex] decreases. For example:
- When [tex]\( x = 0 \)[/tex], [tex]\( \left(\frac{3}{4}\right)^0 = 1 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x \to 0 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x\)[/tex] grows without bound since [tex]\(\left(\frac{3}{4}\right)^{-x} = \left(\frac{4}{3}\right)^x\)[/tex].
2. Shifting the Function Downward:
- The given function is shifted downward by 4 units. So, [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex].
- This shift affects the range of the function. We subtract 4 from each value [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] can take.
3. Finding the Range:
- The exponential function [tex]\( \left(\frac{3}{4}\right)^x \)[/tex] can vary from 0 (exclusive) to 1 (inclusive):
- When [tex]\( x \to \infty \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x \to 0\)[/tex], so [tex]\( \left(\frac{3}{4}\right)^x - 4 \to -4 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\(\left(\frac{3}{4}\right)^x = 1\)[/tex], so [tex]\( f(x) = 1 - 4 = -3\)[/tex].
- Therefore, the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex] gets arbitrarily close to [tex]\(-4\)[/tex] but never actually reaches it, and it can grow without bound as [tex]\( x \)[/tex] moves towards [tex]\(-\infty \)[/tex].
4. Conclusion:
- The range of the function [tex]\( f(x) = \left(\frac{3}{4}\right)^x - 4 \)[/tex] is all real numbers greater than [tex]\(-4\)[/tex]. This can be written in the set notation as [tex]\(\{y \mid y > -4\}\)[/tex].
Given these observations, the correct range of the function is:
[tex]\[ \{y \mid y > -4\} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\{y \mid y > -4\}} \][/tex]
