Use the given function f to answer parts (a) through (f).
$f(x)=\log (x+3)+4$
The given function is \(f(x) = \log(x + 3) + 4\), where \(\log\) denotes the common logarithm (base 10). This function is a transformation of the parent logarithmic function \(g(x) = \log(x)\). Specifically, it represents a horizontal shift to the left by 3 units and a vertical shift upward by 4 units. We will analyze the properties of this function by examining its domain, range, asymptotes, intercepts, and its inverse.
The logarithm \(\log(u)\) is defined only for \(u > 0\). For our function, this requires \(x + 3 > 0\), which simplifies to \(x > -3\). Thus, the domain of \(f(x)\) is the interval \((-3, \infty)\). Regarding the range, as the argument \(x+3\) approaches 0 from the right, the function value approaches \(-\infty\), and as \(x\) increases without bound, \(\log(x+3)\) also increases without bound. Therefore, the range of the function is all real numbers, \((-\infty, \infty)\).
A vertical asymptote occurs where the argument of the logarithm is zero. Setting the argument \(x + 3 = 0\), we find the vertical asymptote at \(x = -3\). For the y-intercept, we evaluate the function at \(x = 0\):
\[f(0) = \log(0 + 3) + 4 = \log(3) + 4.\]
Using the value of the common logarithm, \(\log(3) \approx 0.4771\), the y-intercept is approximately \(4.4771\). For the x-intercept, we set \(f(x) = 0\) and solve for \(x\):
\[\log(x + 3) + 4 = 0\]
\[\log(x + 3) = -4\]
Converting from logarithmic to exponential form:
\[x + 3 = 10^{-4} = 0.0001\]
\[x = 0.0001 - 3 = -2.9999.\]
Thus, the x-intercept is \(x = -2.9999\).
To find the inverse function \(f^{-1}(x)\), we set \(y = f(x)\), swap \(x\) and \(y\), and solve for \(y\):
\[x = \log(y + 3) + 4\]
Subtract 4 from both sides:
\[x - 4 = \log(y + 3)\]
Exponentiate both sides with base 10:
\[10^{x - 4} = y + 3\]
Solve for \(y\):
\[y = 10^{x - 4} - 3\]
Thus, the inverse function is \(f^{-1}(x) = 10^{x - 4} - 3\).
Logarithmic functions are used extensively in fields like chemistry to measure acidity. The pH of a solution is defined as \(pH = -\log[H^+]\), where \([H^+]\) is the concentration of hydrogen ions. Understanding how to shift and transform these logarithmic scales allows scientists to model different chemical behaviors accurately. Similarly, in acoustics, decibel levels are calculated using logarithmic scales, helping engineers design systems for noise control.