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To find the inverse of a function, we need to replace f(x) for y and switch every x for a y, and every y for a x:
[tex]\begin{gathered} f(x)=5^{x+2} \\ y=5^{x+2} \\ x=5^{y+2} \\ \ln x=\ln 5^{y+2} \\ By\text{ properties of logarithms:} \\ \ln (x)=(y+2)\ln (5) \\ \frac{\ln(x)}{\ln(5)}=y+2 \\ y=\frac{\ln(x)}{\ln(5)}-2 \\ f^{-1}(x)=\frac{\ln(x)}{\ln(5)}-2 \end{gathered}[/tex]
1. The graph of f^-1(x) would be:
2. Domain of a function is all the set x-values or input values of a function, so in this case:
As we can see in the graph the function goes from (0, ∞), then its domain:
[tex]D_{f^{-1}(x)}=(0,\text{ }\infty)^{}[/tex]
3. Range is the set of y-values that the function can take or output values, in this case, we can see it goes from 0 to -∞, then its range would be:
[tex]R_{f^{-1}(x)}=(0,-\infty)[/tex]
4. In the graph, we can see that from 0 to ∞, the function is increasing.
5. Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function, we can see that the asymptote would be x=0.
