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An exponential function is a type of mathematical function of the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent.In our exercise, the exponential function is \( f(x) = 4^x \). This function grows rapidly as \( x \) increases. For example:

• When \( x = 0 \), \( 4^0 = 1 \).
• When \( x = 1 \), \( 4^1 = 4 \).
• When \( x = 2 \), \( 4^2 = 16 \).

As you can see, as the exponent \( x \) increases, the value of \( 4^x \) increases very quickly. This is what we call exponential growth.Exponential functions have a horizontal asymptote at \( y = 0 \), meaning \( y \) gets very close to zero as \( x \) becomes increasingly negative.

The inverse function essentially reverses the process of the original function. For the exponential function \( f(x) = 4^x \), the inverse function is \( f^{-1}(x) = \log_{4} x \).The inverse function, also known as the logarithmic function, tells you the exponent to which the base (in this case, 4) must be raised to obtain a given number.For example:

• \( \log_{4}(1) = 0 \) because \( 4^0 = 1 \).
• \( \log_{4}(4) = 1 \) because \( 4^1 = 4 \).
• \( \log_{4}(16) = 2 \) because \( 4^2 = 16 \).

So, while the exponential function rapidly grows, the logarithmic function slowly increases over time. They are mirror images when plotted on a coordinate plane with the line \( y = x \) in between them.

A logarithmic function is the inverse of an exponential function. For our case, \( f^{-1}(x) = \log_{4} x \) tells how many times we need to multiply 4 by itself to get \( x \). It effectively 'undoes' what the exponential function does.Logarithmic functions grow more slowly compared to exponential functions. For instance,

• With \( x = 1 \), \( \log_{4}(1) = 0 \).
• With \( x = 4 \), \( \log_{4}(4) = 1 \).
• With \( x = 16 \), \( \log_{4}(16) = 2 \).

These values make sense because exponentiating 4 gives the values found inside the logarithmic function.The logarithmic curve will intersect the y-axis at a point reflecting this gradual growth, forming a perfect mirror to the exponential function around the line \( y = x \).

The coordinate plane is a fundamental concept for graphing functions. It consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).Each point on the coordinate plane is defined by a pair of numerical coordinates \( (x, y) \). These coordinates help us locate points and graph curves.To plot the functions \( f(x) = 4^x \) and \( f^{-1}(x) = \log_{4} x \), we choose various values for \( x \) and compute the corresponding \( y \).For example:

• For \( f(x) = 4^x \), plotting the points \( (-2, \frac{1}{16}) \), \( (-1, \frac{1}{4}) \), \( (0, 1) \), \( (1, 4) \), and \( (2, 16) \) will help shape the exponential curve.
• Conversely, for \( f^{-1}(x) = \log_{4} x \), plotting points like \( (\frac{1}{16}, -2) \), \( (\frac{1}{4}, -1) \), \( (1, 0) \), \( (4, 1) \), and \( (16, 2) \) will shape the logarithmic curve.

Using these coordinates, we can visually interpret both functions on one set of axes.

A useful concept when dealing with inverse functions is the reflection over the line \( y = x \).If you have a function and its inverse, they will be symmetrical about the line \( y = x \).In our exercise, plotting \( f(x) = 4^x \) and \( f^{-1}(x) = \log_{4} x \) illustrates this reflection.For each point \( (a, b) \) on the exponential function's graph, there will be a corresponding point \( (b, a) \) on the logarithmic function's graph.Drawing the line \( y = x \) helps see this symmetry more clearly.Graphing both functions on the same coordinate plane and adding the line \( y = x \) shows how each function is a mirror image of the other on either side of this line.Understanding this reflection property simplifies comprehending their inverse nature.