Question

(Remember to show work and explain)

Find the inverse of the functions.


`y = {e}^(x - 4)`

`y = {5}^(x) - 9`

`y = 13 + log \: x`
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Answer 1

y = lnx + 4.

y = log 5 (x + 9).

y = 10^(x-13).

Explanation:

To find the inverse you need to make x the subject of the equation:

y = e^(x - 4)

By the definition of a logarithm:

x - 4 = ln y

x = ln y + 4

Now swap x's and y's :-

The inverse is y = lnx + 4.

y = 5^x - 9

Swap x and y:

x = 5^y - 9

5^y = x + 9

The inverse is y = log5( x + 9).

y = 13 + log x

log x = y - 13

x = 10^(y-13)

The inverse is y = 10^(x-13).

Answer 2

Answer:
`\bold{f^(-1)(x)=ln(x)+4}`

Explanation:

Find the inverse by swapping the x's and y's and then solving for y.

`y=e^(x-4)\\\\\\\text{swap the x and y}:\\x=e^(y-4)\\\\\\\text{apply ln to both sides (to eliminate e)}:\\ln(x)=ln(e^(y-4))\quad \rightarrow \quad ln(x)=y-4\\\\\\\text{add 4 to both sides}:\\ln(x)+4=y\\\\\\\text{Therefore, the inverse is: }f^(-1)(x)=ln(x)+4`

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Answer:
`\bold{f^(-1)(x)=log_5(x+9)}`

Explanation:

Find the inverse by swapping the x's and y's and then solving for y.

`y=5^x-9\\\\\\\text{swap the x and y}:\\x=5^y-9\\\\\\\text{add 9 to both sides}:\\x+9=5^y\\\\\\\text{Use the exponent-to-log conversion rule (aka apply }log_5\ \text{to both sides)}:\\log_5(x+9)=y\\\\\\\text{Therefore, the inverse is: }f^(-1)(x)=log_5(x+9)`

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Answer:
`\bold{f^(-1)(x)=10^(x-13)}`

Explanation:

Find the inverse by swapping the x's and y's and then solving for y.

`y=13+log(x)\\\\\\\text{swap the x and y}:\\x=13+log(y)\\\\\\\text{subtract 13 from both sides}:\\x-13=log(y)\\\\\\\text{Use the log-to-exponent conversion rule (note: log = }log_(10))\\10^(x-13)=y\\\\\\\text{Therefore, the inverse is: }f^(-1)(x)=10^(x-13)`