Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$f(x)=\frac{x^{2}+4 x-21}{x^{2}-4 x+3}$$

Step-by-Step Solution

Answer

The zeros of \(f(x)\) are x=-7 and x=3. x=1 is excluded as it creates an undefined point in the function.

Step 1: Identify Numerator of the Function

Firstly, recognize the numerator of the function \(f(x)\). Here, the numerator is \(x^{2}+4x-21\). Set this equal to zero, because a fraction is zero only when its numerator is zero.

Step 2: Solve for x

Next, solve the equation \(x^{2}+4x-21=0\) for x. This can be done by factoring the quadratic equation as \((x+7)(x-3)=0\). Then setting each factor equal to zero gives: \(x+7=0\) (which solves to x=-7) and \(x-3=0\) (which solves to x=3). So the zeros of the function are x=-7 and x=3.

Step 3: Check for Restrictions

Now, let's identify any restrictions or the x-values that make the denominator zero. Here, the denominator is \(x^{2}-4x+3\). Set this equal to zero and solve for x. The equation is factored as \((x-1)(x-3)=0\), which gives x=1 and x=3. However, x=3 has already been included as a zero from the numerator result. Therefore, x=1 is excluded since it would make the function undefined.

Step 4: Confirm with Graphing Utility

Finally, verify the solution using a graphing utility. Plot the function \(f(x)=(x^{2}+4 x-21)/(x^{2}-4x+3)\). The x-intercepts (the points where the graph crosses the x-axis) will confirm that the zeros found are correct.

Step 5: Conclusion

The zeros of the function \(f(x)\) are x=-7 and x=3, excluding x=1 due to an undefined point. This was confirmed using a graphing utility.

Key Concepts

Graphing Rational Functions

When graphing rational functions like the one given by the equation \(f(x)=\frac{x^2+4x-21}{x^2-4x+3}\), there are a few key steps to follow. First, you need to find the x-intercepts by setting the numerator equal to zero and solving for x. In this case, factoring the numerator gives us two potential zeros. Next, it's crucial to locate any vertical asymptotes by finding the zeros of the denominator, which indicate the values for which the function is undefined. Additionally, horizontal asymptotes can be determined by the degrees of the numerator and the denominator polynomials. Once these components are identified, sketching the graph involves plotting the intercepts and asymptotes, then drawing the curve of the function while noting the behavior as it approaches the asymptotes. Employing a graphing utility at this point can help to verify the accuracy of your sketched graph and ensure you haven't missed any critical points.

Factor Quadratic Equations

To find the zeros of a rational function, one must first be able to factor the quadratic equations that appear in the numerator and denominator. Factoring quadratic equations is essentially finding two binomials that, when multiplied together, return the original quadratic equation. For example, the quadratic equation \(x^2+4x-21\) can be factored into \(x+7)(x-3)\). Factoring is done by looking for two numbers that multiply to give the constant term (in this case, -21) and add up to the linear coefficient (in this case, 4). Recognizing patterns and practicing various factoring techniques like the difference of squares, perfect square trinomials, and others are essential keys to mastering this skill, which is fundamental to working with rational functions.

X-Intercepts of Functions

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. To find them, we set the y-value to zero and solve for x. In the context of rational functions, the x-intercepts are found by setting the numerator equal to zero because a fraction is only equal to zero when its numerator is zero. For our given function \(f(x)\), the x-intercepts are at \(x=-7\) and \(x=3\), as found by factoring the numerator and solving for x. It's important to also consider any common factors with the denominator at this point, because if one of these intercepts also makes the denominator zero, it is not considered an x-intercept but an undefined point in the function.

Undefined Points in Functions

Undefined points in functions occur wherever the function cannot produce a value. In rational functions, these undefined points happen where the denominator is equal to zero since division by zero is undefined in mathematics. In our given function \(f(x)\), setting the denominator \(x^2-4x+3\) equal to zero and factoring results in two values, x=1 and x=3. It's critical to exclude these points from the graph of the function, as they represent vertical asymptotes, rather than points on the curve. The process of determining the undefined points protects against misinterpreting these values as valid solutions when finding zeros, intercepts, or sketching the graph.