To understand the effect of the transformation on the function [tex]\( f(x) = 2^x \)[/tex] when it becomes [tex]\( g(x) = 2^{-x} \)[/tex], let's carefully analyze what happens step by step.
1. Original Function:
The original function is given by [tex]\( f(x) = 2^x \)[/tex]. This is an exponential function where the base is 2 and the exponent is [tex]\( x \)[/tex]. This function grows rapidly as [tex]\( x \)[/tex] increases because the base 2 is greater than 1.
2. Transformation to [tex]\( g(x) \)[/tex]:
The transformed function is [tex]\( g(x) = 2^{-x} \)[/tex]. We can rewrite [tex]\( g(x) \)[/tex] to make the transformation more apparent:
[tex]\[ g(x) = 2^{-x} = \left(2^{-1}\right)^x = \left(\frac{1}{2}\right)^x \][/tex]
Here, we see that the exponent [tex]\( x \)[/tex] now applies to the base [tex]\(\frac{1}{2}\)[/tex] instead of 2. The function [tex]\( \left(\frac{1}{2}\right)^x \)[/tex] decreases as [tex]\( x \)[/tex] increases because the base [tex]\(\frac{1}{2}\)[/tex] is less than 1.
3. Effect of the Transformation:
To understand the geometric effect on the graph of [tex]\( f(x) \)[/tex], consider what happens when you replace [tex]\( x \)[/tex] with [tex]\(-x\)[/tex] in [tex]\( f(x) = 2^x \)[/tex].
For any point [tex]\((x, 2^x)\)[/tex] on the graph of [tex]\( f(x)\)[/tex], the corresponding point on the graph of [tex]\( g(x) \)[/tex] is [tex]\((x, 2^{-x})\)[/tex]. Notice that:
[tex]\[ g(x) = 2^{-x} = f(-x) \][/tex]
This means for every [tex]\( x \)[/tex], the function value [tex]\( g(x) \)[/tex] is the same as [tex]\( f(-x) \)[/tex]. Geometrically, this corresponds to reflecting every point [tex]\((x, f(x))\)[/tex] over the y-axis to [tex]\((-x, f(x))\)[/tex].
4. Conclusion:
The transformation from [tex]\( f(x) = 2^x \)[/tex] to [tex]\( g(x) = 2^{-x} \)[/tex] results in reflecting the graph of [tex]\( f(x) \)[/tex] across the y-axis. Hence, the effect of the transformation is a reflection across the y-axis.
So, the final statement is:
"The effect of the transformation on [tex]\( f(x) = 2^x \)[/tex] to become [tex]\( g(x) = 2^{-x} \)[/tex] is a reflection across the y-axis."
