Chapter 9.2, Problem 10QR

To determine

To Write: A function in $x$ as a function of $h\left(=\frac{1}{x}\right)$

Answer to Problem 10QR

The function transformed into the variable $h$ is $y={\left(1+h\right)}^{\frac{1}{h}}$

Explanation of Solution

Given Information: The function is $y={\left(1+\frac{1}{x}\right)}^x$

Method: Manipulate the expression algebraically to write it in terms of $\frac{1}{x}$ and then replace $\frac{1}{x}$ with $h$ .

Apply logarithm on both sides of the equation of the function to write

  $\ln y=\ln \left[{\left(1+\frac{1}{x}\right)}^x\right]=x\ln \left(1+\frac{1}{x}\right)$

Now multiply and divide the righthand side of the equation with $\frac{1}{x}$ to write

  $\ln y=\frac{\frac{1}{x}\cdot xln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}=\frac{ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}$

Nowreplace $\frac{1}{x}$ with $h$ to write

  $\ln y=\frac{\ln (1+h)}{h}=\ln \left[{\left(1+h\right)}^{\frac{1}{h}}\right]$

  $\Rightarrow y=(1+h{)}^{\frac{1}{h}}$