Problem 96 - College Algebra Essentials

Question

Show that $$f(x)=\frac{3 x-2}{5 x-3}$$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{3x-2}{5x-3}\) is a valid function, but does not exist at x = 3/5 due to a point of discontinuity. The graph of this function behaves like a straight line, except for at x = 3/5 where it is undefined.

Step 1: Understanding the function

The function \(f(x)=\frac{3x-2}{5x-3}\) is a ratio of two linear equations, which makes it a rational function. It is important to know the behavior of this function to show its properties.

Step 2: Identify where the function is undefined

The function will be undefined when the denominator equals zero. So first, solve for \(x\) in the equation \(5x-3 = 0\). This gives \(x=\frac{3}{5}\). This is the point of discontinuity.

Step 3: Plotting the function

Plot the function taking note of where x = 3/5. The function behaves like a straight line except it doesn't exist at x = 3/5. The values on both sides of 3/5 will be close, but there will be a gap at x = 3/5. This graphical representation will provide a visual confirmation of the properties and behavior of the function.

Key Concepts

Discontinuities in Functions

A discontinuity in a function occurs at a point where the function is not defined or does not have a value. In the context of rational functions, like
\(f(x) = \frac{3x - 2}{5x - 3}\),
discontinuities often occur when the denominator is zero, since division by zero is undefined. To find discontinuities, we set the denominator equal to zero and solve for \(x\). For our example, setting \(5x - 3 = 0\) and solving gives us \(x = \frac{3}{5}\).

Rational functions can have different types of discontinuities, mainly:

• Point Discontinuity: This occurs when a function is undefined at a particular point, but is defined immediately on either side of that point.
• Jump Discontinuity: This happens when a function suddenly jumps from one value to another as we move along the \(x\)-axis.
• Infinite Discontinuity: Where the function approaches infinity near the discontinuity.

In our function, the type of discontinuity at \(x = \frac{3}{5}\) can be confirmed by looking at a graph of the function or by analyzing the limits of the function as \(x\) approaches \(\frac{3}{5}\) from the left and the right.

Graphing Rational Functions

Graphing rational functions is a skill that combines a solid understanding of algebra with aspects of analytical geometry. The function given,
\(f(x) = \frac{3x - 2}{5x - 3}\),
illustrates common features of rational functions. To graph it, we follow these steps:

• Identify and mark the discontinuities. As calculated earlier, \(x = \frac{3}{5}\) is a value at which the graph will have a hole or a break.
• Find the \(x\)- and \(y\)-intercepts. The \(x\)-intercept is where the graph crosses the \(x\)-axis (set \(f(x) = 0\)), and the \(y\)-intercept is where the graph crosses the \(y\)-axis (set \(x = 0\)).
• Check for horizontal or slant asymptotes which occur as \(x\) approaches infinity or negative infinity. For the given function, you can compare the degrees of the polynomial in the numerator and denominator to determine if a horizontal asymptote exists.
• Plot additional points if necessary for accuracy and draw the curve, paying special attention around the discontinuity.

After plotting, we notice the curve's behavior: approaching the discontinuity, the function values get very large or very small, indicating where the graph will have a vertical asymptote. On either side of the discontinuity, the graph will closely resemble a line but with a gap where \(x\) equals \(\frac{3}{5}\).

Linear Equations

Linear equations are the foundation upon which more complicated functions, like rational functions, are built. They are equations of the first degree, meaning they have variables raised to no higher than the first power. An example is the equation \(5x - 3 = 0\), from our step-by-step solution.

Here are some characteristics of linear equations:

• They graph as straight lines.
• They have a constant rate of change, which is the slope of the line.
• Linear equations have the form \(y = mx + b\) where \(m\) is the slope and \(b\) the \(y\)-intercept.

The equations \(3x - 2\) and \(5x - 3\) in the function \(f(x) = \frac{3x - 2}{5x - 3}\) are both linear. The graph of \(f(x)\) will most closely resemble the line described by the numerator but with modifications by the denominator, including the vertical asymptote and potential horizontal or slant asymptote, influenced by the ratio of the degrees of the polynomials in the numerator and the denominator.