The question asks about the transformation effect on the function [tex]\( f(x) = 2^x \)[/tex] when it becomes [tex]\( g(x) = 2^{-x} \)[/tex].
To understand the transformation, let's analyze the changes step-by-step:
1. Identify the original function and the transformed function:
- Original function: [tex]\( f(x) = 2^x \)[/tex]
- Transformed function: [tex]\( g(x) = 2^{-x} \)[/tex]
2. Understanding the transformation:
- In the transformed function [tex]\( g(x) = 2^{-x} \)[/tex], the exponent [tex]\( x \)[/tex] in the original function [tex]\( f(x) = 2^x \)[/tex] is replaced by [tex]\( -x \)[/tex].
3. Effect of changing [tex]\( x \)[/tex] to [tex]\( -x \)[/tex]:
- Changing [tex]\( x \)[/tex] to [tex]\( -x \)[/tex] in the exponent effectively reflects the function over the [tex]\( y \)[/tex]-axis.
- Why? Reflection over the [tex]\( y \)[/tex]-axis means that if a point [tex]\( (x, y) \)[/tex] is on the graph of [tex]\( f(x) \)[/tex], then the point [tex]\( (-x, y) \)[/tex] will be on the graph of [tex]\( g(x) \)[/tex].
4. Illustrate the reflection:
- For example:
- If [tex]\( x = 1 \)[/tex], then [tex]\( f(1) = 2^1 = 2 \)[/tex].
- In [tex]\( g(x) = 2^{-x} \)[/tex], [tex]\( g(1) = 2^{-1} = \frac{1}{2} \)[/tex].
- If [tex]\( x = -1 \)[/tex], then [tex]\( f(-1) = 2^{-1} = \frac{1}{2} \)[/tex].
- For [tex]\( g(x) = 2^{-x} \)[/tex], [tex]\( g(-1) = 2^{1} = 2 \)[/tex].
Therefore, the effect of the transformation on [tex]\( f(x) = 2^x \)[/tex] to become [tex]\( g(x) = 2^{-x} \)[/tex] is that [tex]\( f(x) \)[/tex] is reflected over the [tex]\( y \)[/tex]-axis.
So, the correct answer is:
[tex]\[ \boxed{f(x)\text{ is reflected over the } y\text{-axis.}} \][/tex]
