91Ó°ÊÓ

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\log _{2}(x+5)\) in interval notation. Solution: The domain of all logarithmic functions is \(x>0\). Interval notation: \((0, \infty)\) This is incorrect. What went wrong?

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\ln |x|\) in interval notation. Solution: since the absolute value eliminates all negative numbers, the domain is the set of all real numbers. Interval notation: \((-\infty, \infty)\) This is incorrect. What went wrong?

Use a graphing utility to graph \(y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} .\) State the domain. Determine whether there are any symmetry and asymptote.

The hyperbolic sine function is defined by \(\sinh x=\frac{e^{x}-e^{-x}}{2}\) Find its inverse function \(\sinh ^{-1} x\).

Determine whether each statement is true or false. The horizontal axis is the horizontal asymptote of the graph of \(y=\ln x\).

The hyperbolic tangent is defined by tanh \(x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\) Find its inverse function \(\tanh ^{-1} x\).

Determine whether each statement is true or false. The graphs of \(y=\log x\) and \(y=\ln x\) have the same \(x\) -intercept (1,0).

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=2^{x}$$

Determine whether each statement is true or false. The graphs of \(y=\log x\) and \(y=\ln x\) have the same vertical asymptote, \(x=0\).

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=4^{x} \cdot 3^{x+1}$$