Question

a) If \(g(x)=\log _{\frac{1}{4}} x,\) state the equation of the inverse, \(g^{-1}(x)\) b) Sketch the graph of \(g(x)\) and its inverse. Identify the following characteristics of the inverse graph: . the domain and range \cdot the \(x\) -intercept, if it exists \cdot the \(y\) -intercept, if it exists \bullet the equations of any asymptotes

Short answer

The inverse of \(g(x) = \log_{\frac{1}{4}} x\) is \(g^{-1}(x) = (\frac{1}{4})^x\). The inverse graph has domain \((-\infty, +\infty)\), range \((0, +\infty)\), no \(x\)-intercept, \(y\)-intercept at (0,1), and a horizontal asymptote at \(y=0\).

Step-by-step solution

- Find the inverse function

To find the inverse function, start by writing the equation of the function: \[ g(x) = \log_{\frac{1}{4}} x \] Replace \(g(x)\) with \(y\): \[ y = \log_{\frac{1}{4}} x \] Now switch \(x\) and \(y\): \[ x = \log_{\frac{1}{4}} y \]Convert the logarithmic equation to its exponential form:\[ y = (\frac{1}{4})^x \]Thus, the inverse function is \[g^{-1}(x) = (\frac{1}{4})^x \]

- Sketch the graph of \(g(x)\)

To sketch the graph of \(g(x) = \log_{\frac{1}{4}} x\), remember that the base \(\frac{1}{4}\) is less than 1, which means the graph will be decreasing. Plot points such as (1,0), (\frac{1}{4}, 1) and (4, -1). Connect the points smoothly to form a decreasing curve.

- Sketch the graph of the inverse function

Sketch the graph of \(g^{-1}(x) = (\frac{1}{4})^x\). Plot points like (0,1), (1,\frac{1}{4}) and (-1, 4). Connect these points to form a curve that is decreasing as \(x\) increases.

- Identify characteristics of the inverse graph

The graph of the inverse function \(g^{-1}(x)\) has the following characteristics: - **Domain:** All real numbers (\(-\infty, +\infty\))- **Range:** (0, +\infty)- **x-intercept:** None- **y-intercept:** (0,1)- **Asymptote:** The horizontal asymptote at \(y = 0\)

Key concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. If you have an exponential function of the form \( y = a^x \), the corresponding logarithmic function would be \( x = \log_a(y) \). Here, \( \log_{\frac{1}{4}}x \) describes a function where the base is \( \frac{1}{4} \). The property of logarithms with a base less than 1 results in a decreasing graph as \( x \) increases. This is the nature of the function we dealt with in the problem.

Exponential Functions

Exponential functions grow or decay at a constant rate per unit interval. In our problem, the function \( g^{-1}(x) = (\frac{1}{4})^x \) is the inverse of our logarithmic function. Since the base (\frac{1}{4}) is less than 1, this is an exponential decay function. The inverse is identified by converting the logarithmic equation \( y = \log_{\frac{1}{4}}x \) to its exponential form, resulting in \( y = (\frac{1}{4})^x \).

Graph Characteristics

The characteristics of the graph of a function and its inverse are crucial to understanding their behavior. For \( g(x) = \log_{\frac{1}{4}}x \): - **Domain:** \( (0, +\infty) \) as logarithms are undefined for non-positive numbers.
- **Range:** \( (-\infty, +\infty) \).
- **Intercepts:** The x-intercept can be found at (1,0).
For its inverse \( g^{-1}(x) = (\frac{1}{4})^x \):

• **Domain:** All real numbers \( ( -\infty, +\infty) \).
• **Range:** (0, +\infty), as exponential functions are never negative.
• **Intercepts:** It has a y-intercept at (0,1).

Asymptotes

Asymptotes are lines that a graph approaches but never reaches. For the exponential function \( g^{-1}(x) = (\frac{1}{4})^x \), it has a horizontal asymptote at \( y = 0 \). This means no matter how large (or small) \( x \) gets, the value of \( (\frac{1}{4})^x \) will approach 0 but never truly be 0. This property is mirrored in logarithmic functions where the vertical asymptote is at \( x = 0 \).