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Simplify $\text{cosh}\left(2\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}x\right).$
$({x}^{2}+{x}^{\mathrm{-2}})\text{/}2$
From the graphs of the hyperbolic functions, we see that all of them are one-to-one except $\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ and $\text{sech}\phantom{\rule{0.1em}{0ex}}x.$ If we restrict the domains of these two functions to the interval $\left[0,\infty \right),$ then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions . Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.
Inverse Hyperbolic Functions
Let’s look at how to derive the first equation. The others follow similarly. Suppose $y={\text{sinh}}^{\mathrm{-1}}x.$ Then, $x=\text{sinh}\phantom{\rule{0.1em}{0ex}}y$ and, by the definition of the hyperbolic sine function, $x=\frac{{e}^{y}-{e}^{\text{\u2212}y}}{2}.$ Therefore,
Multiplying this equation by ${e}^{y},$ we obtain
This can be solved like a quadratic equation, with the solution
Since ${e}^{y}>0,$ the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that
Evaluate each of the following expressions.
${\text{sinh}}^{\mathrm{-1}}\left(2\right)=\text{ln}\left(2+\sqrt{{2}^{2}+1}\right)=\text{ln}\left(2+\sqrt{5}\right)\approx 1.4436$
${\text{tanh}}^{\mathrm{-1}}(1\text{/}4)=\frac{1}{2}\text{ln}\left(\frac{1+1\text{/}4}{1-1\text{/}4}\right)=\frac{1}{2}\text{ln}\left(\frac{5\text{/}4}{3\text{/}4}\right)=\frac{1}{2}\text{ln}\left(\frac{5}{3}\right)\approx 0.2554$
Evaluate ${\text{tanh}}^{\mathrm{-1}}(1\text{/}2).$
$\frac{1}{2}\text{ln}\left(3\right)\approx 0.5493.$
For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.
$f\left(x\right)={5}^{x}$ a. $x=3$ b. $x=\frac{1}{2}$ c. $x=\sqrt{2}$
a. 125 b. 2.24 c. 9.74
$f\left(x\right)={\left(0.3\right)}^{x}$ a. $x=\mathrm{-1}$ b. $x=4$ c. $x=\mathrm{-1.5}$
$f\left(x\right)={10}^{x}$ a. $x=\mathrm{-2}$ b. $x=4$ c. $x=\frac{5}{3}$
a. 0.01 b. 10,000 c. 46.42
$f\left(x\right)={e}^{x}$ a. $x=2$ b. $x=\mathrm{-3.2}$ c. $x=\pi $
For the following exercises, match the exponential equation to the correct graph.
For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.
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