Problem 46 - Algebra and Trigonometry

Question

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(f(x)=(1/4)^{x}\) is a decreasing exponential curve. The function \(g(x)=\log_{4}x\) is a logarithmic curve that increases more slowly as x gets larger. These functions are inverses of each other and reflect each other across the line \(y=x\).

Step 1 - Graph of \(f(x)=(1/4)^{x}\)

To graph the function \(f(x)=(1/4)^{x}\), it is essential to understand that this is an exponential function and its graph falls from left to right. Draw an x-y plane and plot the points: when \(x=0\), \(f(x)=1\); when \(x=1\), \(f(x)=1/4\); and when \(x=-1\), \(f(x)=4\). Connect these points smoothly to get the graph of this function.

Step 2 - Graph of \(g(x)=\log_{4}x\)

To graph the function \(g(x)=\log_{4}x\), keep in mind that it is a logarithmic function. On your x-y plane, plot points: when \(x=1\), \(g(x)=0\); when \(x=4\), \(g(x)=1\); and when \(x=1/4\), \(g(x)=-1\). Smoothly connect these points to draw the graph of this function.

Step 3 - Show the reflection

Both graphs should be plotted on the same coordinate system. You will see that the logarithmic function is a reflection of the exponential function in the line \(y=x\) because these two functions are inversely related to each other. This means that a point on one graph at (a,b) will have a corresponding point at (b,a) on the other graph.

Key Concepts

Understanding Exponential Functions

Exponential functions are mathematical expressions in which a variable appears in the exponent. The general form is \( f(x) = a^x \), where \( a \) is a constant. In the function \( f(x) = \left(\frac{1}{4}\right)^x \), the base is \( \frac{1}{4} \), making it a decreasing exponential function.

As \( x \) increases, \( f(x) \) moves closer to zero, causing it to fall from left to right when plotted on a graph. A few key points to consider when graphing:

• At \( x = 0 \), \( f(x) = 1 \); this is the y-intercept.
• For \( x = 1 \), the value is \( f(x) = \frac{1}{4} \).
• For \( x = -1 \), the function "inverts," yielding \( f(x) = 4 \).

When you connect these points on an x-y plane, you'll see a smooth curve reflecting how the function decreases as \( x \) increases. Always remember that exponential functions have horizontal asymptotes, typically along the x-axis in this form.

Insight into Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they are defined as \( g(x) = \log_b x \), where \( b \) is the base. For the function \( g(x) = \log_4 x \), the base is 4, making this a logarithmic function with a gentle upward slope.

Logarithmic functions increase slowly and have a vertical asymptote. Here's how you plot it:

• For \( x = 1 \), the output is \( g(x) = 0 \), defining the x-intercept.
• For \( x = 4 \), \( g(x) = 1 \).
• When \( x = \frac{1}{4} \), the function dips to \( g(x) = -1 \).

These points, once connected, show how the logarithmic curve trends upward. Logarithmic graphs usually never touch or cross the y-axis because \( x \) cannot be zero or negative.

Exploring Inverse Functions

Inverse functions essentially "reverse" each other's effects. For instance, if an exponential function raises a number by an exponent, a logarithmic function finds the original number based on another's exponentiation.

When graphing the exponential function \( f(x) = \left(\frac{1}{4}\right)^x \) and the logarithmic function \( g(x) = \log_4 x \) together, you'll notice that they reflect along the line \( y = x \). This means any point on one graph corresponds to a reversed point on the other graph.

• If a point on the graph of \( f(x) \) is \((a, b)\), the corresponding point on \( g(x) \) will be \((b, a)\).
• This mirrored arrangement underlines their inverse relationship.

Understanding their reflective nature is key to identifying inverse function pairs and their unique behavior on a coordinate plane. This symmetry shows how inversely related functions interact graphically.