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 $$ \cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y $$ for all real numbers \(x\) and \(y\).

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=8 \cdot 7^{x} $$

Explain why $$ 1+\log x=\log (10 x) $$ for every positive number \(x\).

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

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 $$ \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y $$ for all real numbers \(x\) and \(y\).

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=3 \cdot 4^{x}-5 $$

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 $$ \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y $$ for all real numbers \(x\) and \(y\).

Explain why $$ 2-\log x=\log \frac{100}{x} $$ for every positive number \(x\).

Explain why $$ (1+\log x)^{2}=\log \left(10 x^{2}\right)+(\log x)^{2} $$ for every positive number \(x\).

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