Math

QuestionHow to prove j(x)=11.6exj(x)=11.6 e^{x} and k(x)=ln(x11.6)k(x)=\ln \left(\frac{x}{11.6}\right) are inverses? Show j(k(x))=xj(k(x))=x.

Studdy Solution

STEP 1

Assumptions1. The function j(x)j(x) is defined as j(x)=11.6exj(x) =11.6 e^{x} . The function k(x)k(x) is defined as k(x)=ln(x11.6)k(x) = \ln \left(\frac{x}{11.6}\right)3. We need to show that j(x)j(x) and k(x)k(x) are inverse functions

STEP 2

To show that two functions are inverses of each other, we need to show that both j(k(x))j(k(x)) and k(j(x))k(j(x)) equal xx.

STEP 3

First, let's calculate j(k(x))j(k(x)). We substitute k(x)k(x) into j(x)j(x).
j(k(x))=11.6ek(x)j(k(x)) =11.6 e^{k(x)}

STEP 4

Now, substitute the definition of k(x)k(x) into the equation.
j(k(x))=11.6eln(x11.6)j(k(x)) =11.6 e^{\ln \left(\frac{x}{11.6}\right)}

STEP 5

We know that elna=ae^{\ln a} = a, so we can simplify the equation.
j(k(x))=11.x11.j(k(x)) =11. \cdot \frac{x}{11.}

STEP 6

implify the equation further.
j(k(x))=xj(k(x)) = x

STEP 7

Now, let's calculate k(j(x))k(j(x)). We substitute j(x)j(x) into k(x)k(x).
k(j(x))=ln(j(x)11.6)k(j(x)) = \ln \left(\frac{j(x)}{11.6}\right)

STEP 8

Now, substitute the definition of j(x)j(x) into the equation.
k(j(x))=ln(11.6ex11.6)k(j(x)) = \ln \left(\frac{11.6 e^{x}}{11.6}\right)

STEP 9

implify the equation.
k(j(x))=ln(ex)k(j(x)) = \ln (e^{x})

STEP 10

We know that lnea=a\ln e^{a} = a, so we can simplify the equation.
k(j(x))=xk(j(x)) = xSince both j(k(x))j(k(x)) and k(j(x))k(j(x)) equal xx, we can conclude that j(x)j(x) and k(x)k(x) are inverse functions.
The correct statement is "It must be shown that both j(k(x))j(k(x)) and k(j(x))k(j(x)) equal xx."

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