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Chapter 1: Problem 29

(a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. \(f(x)=\frac{3 x-1}{x-6}\)

Short Answer

Expert verified
From the graph, the function \(f(x)=\frac{3 x-1}{x-6}\) seems to intersect the x-axis at \(x = 1/3\). Algebraically, \(3x = 1\) gives \(x = 1/3\), confirming the result from the graph.

Step by step solution

01

Graph the function

Using a graphing utility, plot the function \(f(x)=\frac{3 x-1}{x-6}\). This function has a vertical asymptote at \(x=6\) because division by zero is undefined. Observe the points where the function intersects the x-axis to find the zeros of the function.
02

Determine the zeros

The zeros of the function are the x-values which make \(f(x) = 0\). These are the points where the graph intersects the x-axis. By examining the graph, identify these points.
03

Verify algebraically

To verify the graphically obtained zeros, algebraically solve \(f(x) = 0\) for \(x\). This is equivalent to solving the equation \(\frac{3x - 1}{x - 6} = 0\)
04

Solve the equation

Multiply both sides of the equation by \(x-6\) to eliminate the denominator. It results in \(3x - 1 = 0\). Now solve this equation for \(x\) to get the zeros algebraically.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zeros of Functions
Finding the zeros of a function is equivalent to determining the input values (x-values) where the function's output is zero. For the function \( f(x) = \frac{3x - 1}{x - 6} \), the zeros occur when the numerator equals zero because a fraction is zero when its numerator is zero (and the denominator is not zero).

Let's set the numerator equal to zero:
  • Equation: \( 3x - 1 = 0 \)
  • Solve for \( x \): Add 1 to both sides to get \( 3x = 1 \)
  • Divide by 3 to isolate \( x \), resulting in \( x = \frac{1}{3} \)
Thus, the graph intersects the x-axis at \( x = \frac{1}{3} \). This is the point where \( f(x) \) equals zero, and it represents the zero of the function. Graphically, you can also verify the zeros by checking where the graph crosses the x-axis.
Vertical Asymptotes
Vertical asymptotes occur in rational functions when the denominator equals zero and the numerator is not zero at that point. They represent values of \( x \) where the function tends to infinity and the graph spikes up or down without bound.

For the given function \( f(x) = \frac{3x - 1}{x - 6} \), a vertical asymptote occurs when:
  • Denominator: \( x - 6 = 0 \)
  • Solve for \( x \): Add 6 to both sides which gives \( x = 6 \)
At \( x = 6 \), \( f(x) \) is undefined, which is visualized as a vertical line where the graph does not touch.

In your graph, you'll notice that the function approaches this line as \( x \) gets closer to 6, either from the left or the right. This behavior helps in identifying limitations in the logical domain of the function.
Algebraic Verification
Algebraic verification entails confirming graphically obtained solutions through mathematical calculations. It is beneficial as it provides a cross-checking method ensuring accuracy and robustness of results. For the function \( f(x) = \frac{3x - 1}{x - 6} \), we need to determine the zeros by solving \( \frac{3x - 1}{x - 6} = 0 \) algebraically.

Steps for verification include:
  • Ignore the denominator initially as it does not affect the condition for zero (as long as it is non-zero).
  • Set the numerator equal to zero: \( 3x - 1 = 0 \)
  • Solve this simple equation: Add 1 to both sides, \( 3x = 1 \)
  • Divide by 3 to isolate \( x \), resulting in \( x = \frac{1}{3} \)
This algebraic solution confirms what is observed graphically, ensuring that \( x = \frac{1}{3} \) is indeed a zero of the function. By addressing rational functions both graphically and algebraically, students can strengthen their comprehension of function behavior and characteristics.

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