Question

Hello, can you help me with this questions?

1). Assume that f(x) is one-to-one. Find f −1. f(x) = x3 − 11

2). Solve the equation for x. (Enter your answers as a comma-separated list. Round your answers to four decimal places.)

7^x = 9^{x − 1}

Answer 1

1) $\large f(x)=x^3-11$

to find inverse put f(x) as y and find equation for x . Then change x to f^-1(x) and y to x to get this

on applying f(x) as y ,it becomes

$\large y=x^3-11$
find equation for x

$\large x^3=y+11$

3 = y +11

now on changing it becomes

$\large \boldsymbol{f^{-1}(x)=\sqrt[3]{x+11}}$

so this is answer

2)

$\large 7^x= 9^{x-1}$

take natural log on both side

$\large ln(7^x)= ln(9^{x-1})$

we know ln (a^b) = b*ln(a) .

so

$\large x*ln(7)= (x-1)ln(9)$

mon rearranging this

$\large \frac{(x-1)}{x}=\frac{ln(7)}{ln(9)}$

$\large 1-\frac{1}{x}=\frac{ln(7)}{ln(9)}$

$\large \frac{1}{x}=1-\frac{ln(7)}{ln(9)}$

$\large x=\frac{1}{1-\frac{ln(7)}{ln(9)}}=8.7429$

so answer is 8.7429