To solve for the function [tex]\( g(x) = \left(\frac{3}{4}\right)^x \)[/tex], let's break down the process step-by-step.
1. Understand the Function:
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = \left(\frac{3}{4}\right)^x \)[/tex].
- This is an exponential function where the base is [tex]\( \frac{3}{4} \)[/tex] and the exponent is [tex]\( x \)[/tex].
2. Evaluating the Function:
- Consider different values of [tex]\( x \)[/tex] to evaluate [tex]\( g(x) \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = \left(\frac{3}{4}\right)^0 = 1
\][/tex]
Any number raised to the power of 0 is 1.
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = \left(\frac{3}{4}\right)^1 = \frac{3}{4}
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = \left(\frac{3}{4}\right)^2 = \frac{9}{16}
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = \left(\frac{3}{4}\right)^{-1} = \frac{4}{3}
\][/tex]
A negative exponent inverts the fraction.
3. General Observation:
- This function [tex]\( g(x) \)[/tex] decreases as [tex]\( x \)[/tex] increases because [tex]\( 0 < \frac{3}{4} < 1 \)[/tex], and when a fraction less than 1 is raised to a power, the result gets smaller.
- For positive [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will be a fraction.
- For negative [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will result in values greater than 1.
4. Applications:
- Exponential functions like [tex]\( g(x) \)[/tex] are used in various real-world applications, such as population decay, radioactive decay, and financial calculations involving depreciation.
In summary, [tex]\( g(x) = \left(\frac{3}{4}\right)^x \)[/tex] is an exponential function that decreases as [tex]\( x \)[/tex] increases. The function is versatile and can be evaluated for various values of [tex]\( x \)[/tex] to understand its behavior.
