91Ó°ÊÓ

The functions cosh and sinh are defined by \(\cosh x=\frac{e^{x}+e^{-x}}{2} \quad\) and \(\quad \sinh x=\frac{e^{x}-e^{-x}}{2}\) for every real number \(x .\) These functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that if \(x\) is very large, then $$ \cosh x \approx \sinh x \approx \frac{e^{x}}{2}. $$

Explain why $$ \frac{1+\log x}{2}=\log \sqrt{10 x} $$ for every positive number \(x\)

The functions cosh and sinh are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \quad \text { and } \quad \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) These functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that the range of sinh is the set of real numbers.

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\log _{3} x $$

The functions cosh and sinh are defined by \(\cosh x=\frac{e^{x}+e^{-x}}{2} \quad\) and \(\quad \sinh x=\frac{e^{x}-e^{-x}}{2}\) for every real number \(x .\) These functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that sinh is a one-to-one function and that its inverse is given by the formula $$ (\sinh )^{-1}(y)=\ln \left(y+\sqrt{y^{2}+1}\right) $$ for every real number \(y\).

Suppose \(f(x)=\log x\) and \(g(x)=\log (1000 x)\). Explain why the graph of \(g\) can be obtained by shifting the graph of \(f\) up 3 units.

The functions cosh and sinh are defined by \(\cosh x=\frac{e^{x}+e^{-x}}{2} \quad\) and \(\quad \sinh x=\frac{e^{x}-e^{-x}}{2}\) for every real number \(x .\) These functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that the range of cosh is the interval \([1, \infty)\).

Suppose \(f(x)=\log _{6} x\) and \(g(x)=\log _{6} \frac{36}{x} .\) Explain why the graph of \(g\) can be obtained by flipping the graph of \(f\) across the horizonal axis and then shifting up 2 units.

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=5+3 \log _{6}(2 x+1) $$

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=8+9 \log _{2}(4 x-7) $$