91Ó°ÊÓ

Know that the hyperbolic cosine function, \(\cosh x\), is defined as: \(\cosh x = \frac{e^x + e^{-x}}{2}\) To sketch the function, we'll plot a few key points: 1. For \(x = 0\), \(\cosh 0 = \frac{e^0 + e^{-0}}{2} = \frac{1 + 1}{2} = 1\) 2. As \(x\) goes to positive or negative infinity, \(\cosh x\) will always be positive and grow exponentially. By plotting these points and noting the properties, we find that the graph of \(y = \cosh x\) has no asymptotes, since it just grows to infinity as \(x\) increases or decreases. Since \(\cosh(-x) = \cosh x\), this function is even.

The hyperbolic sine function, \(\sinh x\), is defined as: \(\sinh x = \frac{e^x - e^{-x}}{2}\) To sketch the function, let's plot a few key points: 1. For \(x = 0\), \(\sinh 0 = \frac{e^0 - e^{-0}}{2} = \frac{1 - 1}{2} = 0\) 2. As \(x\) grows large, \(\sinh x\) grows exponentially. 3. As \(x\) goes to negative infinity, \(\sinh x\) becomes increasingly negative. By plotting these points and noted properties, we can see that the graph of \(y = \sinh x\) has no asymptotes and the graph will resemble an exponential function that spans across the \(x\)-axis. Since \(\sinh(-x) = -\sinh x\), this function is odd.

The hyperbolic tangent function, \(\tanh x\), is defined as: \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\) To sketch the function, let's plot a few key points and properties: 1. For \(x = 0\), \(\tanh 0 = 0.\) 2. As \(x\) increases, the function approaches the horizontal asymptote \(y = 1\). 3. As \(x\) decreases, the function approaches the horizontal asymptote \(y = -1\). 4. The function \(\tanh x\) is continuous and smooth and passes through the origin. By plotting these points and noting the asymptotes and properties, we have the graph of the hyperbolic tangent function. Since \(\tanh(-x) = -\tanh x\), this function is also odd. In conclusion, the functions are: - \(y = \cosh x\): Even function, no asymptotes. - \(y = \sinh x\): Odd function, no asymptotes. - \(y = \tanh x\): Odd function, horizontal asymptotes at \(y = 1\) and \(y = -1\).