Problem 60 - College Algebra

Question

Graph \(y-3^{x}\) and \(x-3^{y}\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph shows how the exponential function \(y=3^{x}\) and its inverse \(y=\log_{3}{x}\) behave. For \(y=3^{x}\), the function increases rapidly for positive x and approaches 0 for negative y. For \(y=\log_{3}{x}\), the function is defined for positive x and reflects the graph of \(y=3^{x}\) through the line y=x.

Step 1: Expressing the Functions in the Base Form

Before graphing, it is helpful to understand the functions in their base forms. The base form of an exponential function is \(y=a^{x}\), where a is a positive real number, not equal to 1. The given functions can be rearranged to this form, resulting in, \(y=3^{x}\) and \(y=\log_{3}{x}\).

Step 2: Plotting \(y=3^{x}\)

This is an exponential function with base 3. For positive x, as x increases, y increases rapidly. For negative x, as x decreases, y approaches 0 but never reaches it. Sketch the graph accordingly.

Step 3: Plotting \(y=\log_{3}{x}\)

This is the inverse of the exponential function with base 3. It can be obtained by reflecting the graph of \(y=3^{x}\) through the line y=x. The function is defined for x>0.

Step 4: Overlaying the Graphs

Once both functions are sketched out, overlay them on the same rectangular coordinate system, allowing a comparison of the properties of the exponential function and its inverse, the logarithmic function.

Key Concepts

Graphing

Graphing functions is a way to visually represent mathematical equations on a coordinate plane. In this case, we are looking at two different types of functions: exponential and logarithmic. First, let's consider the function \( y = 3^{x} \). This is an exponential function characterized by the base number 3. As you graph it:

• For positive values of \( x \), \( y \) increases very quickly, producing a curve that rises steeply to the right.
• For negative values of \( x \), the graph approaches the x-axis but never touches it, illustrating that \( y \) gets closer to zero.

This is because exponential functions have a horizontal asymptote, a line that the graph approaches but never quite reaches.Comparatively, plotting the inverse function \( y = \log_{3}{x} \) starts from the opposite perspective. Here, the graph only exists for \( x > 0 \) and represents how many times 3 must be multiplied by itself to get \( x \). This markedly different shape must be placed on the same graph to compare the exponential growth with the logarithmic increase.

Inverse Functions

Inverse functions essentially "undo" each other. When you have a function and its inverse, applying both in sequence will get you back to the original input. In this scenario:

• The function \( y = 3^{x} \) represents exponential growth.
• To find its inverse, we need a function that can reverse its effect: \( y = \log_{3}{x} \).

Visually on a graph, the inverse relationship reflects through the line \( y = x \). When you plot the graphs of an exponential function and its logarithmic inverse, the curve of one function is a mirror image of the other with respect to this line. This symmetry showcases their inverse nature, highlighting how they are mathematically interconnected, with no value being repeated in their mappings.

Logarithmic Functions

Logarithmic functions are derived from exponential functions, and they have unique properties. For an exponential function \( y = 3^{x} \), the logarithm base 3 function is \( y = \log_{3}{x} \). Understanding this:

• A logarithm answers the question: "To what exponent must the base be raised, to yield x?"
• Its graph has the characteristic curve: it starts from negative infinity heading upwards and gradually approaches both the y-axis and to the right, never touching the y-axis.
• The function is only defined for \( x > 0 \), which is why its graph does not exist on the negative side of the x-axis.

The domain of logarithmic functions is limited compared to exponential ones, which shows that while exponentials can cover all real numbers for \( x \), logarithms start from an undefined point at zero and extend to infinity. This limitation is crucial when solving real-world problems involving growth and scale.