Answer
g(x) = 0.5(4)–x and g(x) =1/2(1/4) x
Explanation
To reflect a function over the y-axis we use the rule:
[tex]f(-x)[/tex] reflects the graph of [tex]f(x)[/tex] over the y-axis. In other words, to reflect [tex]f(x)[/tex] over the y-axis, we just need to replace [tex]x[/tex] with [tex]-x[/tex].
Let's find [tex]f(x)[/tex] first using the standard exponential growth function: [tex]f(x)=a(1+b)^x[/tex]
From our graph we can infer that [tex]f(x)=\frac{1}{2}[/tex] when [tex]x=0[/tex], so let's replace those values in our function:
[tex]f(x)=a(1+b)^x[/tex]
[tex]\frac{1}{2} =a(1+b)^0[/tex]
[tex]\frac{1}{2} =a(1)[/tex]
[tex]\frac{1}{2} =a[/tex]
We can also infer that [tex]f(x)=8[/tex] when [tex]x=2[/tex]. Since we already know that [tex]a=\frac{1}{2}[/tex], let's replace the values to find [tex]b[/tex] an complete our function:
[tex]f(x)=a(1+b)^x[/tex]
[tex]8=\frac{1}{2} (1+b)^2[/tex]
[tex]16=(1+b)^2[/tex]
[tex]\sqrt{16} =1+b[/tex]
[tex]4=1+b[/tex]
[tex]b=3[/tex]
So putting it all together:
[tex]f(x)=a(1+b)^x[/tex]
[tex]f(x)=\frac{1}{2} (1+3)^x[/tex]
[tex]f(x)=\frac{1}{2} (4)^x[/tex]
Finally, we can apply the reflection rule to find the reflection of [tex]f(x)[/tex] over the y-axis:
[tex]f(-x)=\frac{1}{2} (4)^{-x}[/tex]
[tex]g(x)=\frac{1}{2} (4)^{-x}[/tex]
Since [tex]\frac{1}{2} =0.5[/tex], [tex]g(x)=\frac{1}{2} (4)^{-x}[/tex] and [tex]g(x)=0.5(4)^{-x}[/tex] are equivalent.
You can also express [tex]g(x)=\frac{1}{2} (4)^{-x}[/tex] as [tex]g(x)=\frac{1}{2} (\frac{1}{4} )^x[/tex] using laws of exponents.
