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Chapter 4: Problem 9

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{x^{2}-9}{x^{2}}\)

Short Answer

Expert verified
The x-intercepts are at \(x=3\) and \(x=-3\); there is no y-intercept.

Step by step solution

01

Find the x-intercepts

The x-intercepts of a function occur where the function equals zero, i.e., where the numerator of the rational function is zero. In this case, we need to solve the equation \(x^2 - 9 = 0\). Factoring the expression, we have \((x-3)(x+3) = 0\). Setting each factor to zero, we find the x-intercepts at \(x = 3\) and \(x = -3\).
02

Find the y-intercept

The y-intercept occurs where \(x = 0\). Plug \(x = 0\) into the function \(r(x) = \frac{x^2 - 9}{x^2}\) to find the y-value. Substitute 0 into the function to get \(r(0) = \frac{0^2 - 9}{0^2}\), which is undefined, meaning there is no y-intercept for this function.
03

Determine domain restrictions

Before concluding, identify restrictions. The denominator \(x^2\) cannot be zero, so \(x eq 0\). This confirms the non-existence of the y-intercept and provides critical values for asymptotic behavior.

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Key Concepts

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

Understanding x-intercepts in Rational Functions
The x-intercepts of a rational function are those values of \( x \) where the function equals zero. In a rational function, which is a fraction of polynomial expressions, the function equals zero when the numerator is zero. Therefore, to find the x-intercepts, we specifically solve the equation where the numerator is zero.
For the rational function given as \( r(x) = \frac{x^2 - 9}{x^2} \), the numerator is \( x^2 - 9 \). Setting this equal to zero, we solve:
  • \( x^2 - 9 = 0 \)
  • Factor it to \( (x - 3)(x + 3) = 0 \)
  • This gives us \( x = 3 \) and \( x = -3 \)
These points, \( x = 3 \) and \( x = -3 \), are where the graph of the function crosses the x-axis, known as the x-intercepts.
Exploring y-intercepts in Rational Functions
The y-intercept of a function is found where the graph crosses the y-axis. This occurs when \( x = 0 \). For a rational function, you substitute \( x = 0 \) into the function to determine the y-intercept. However, in some cases, like with the function \( r(x) = \frac{x^2 - 9}{x^2} \), the situation is different.
When substituting \( x = 0 \) in this function, we see that:
  • \( r(0) = \frac{0^2 - 9}{0^2} \)
  • This becomes \( \frac{-9}{0} \), which is undefined.
Since division by zero is not possible, there is no y-intercept for this rational function. This means the graph does not intersect the y-axis, highlighting an important characteristic of this function when evaluated at \( x = 0 \).
Domain Restrictions in Rational Functions
Domain restrictions determine the set of permissible \( x \) values for a function. These are critical in rational functions due to their denominators. A rational function is undefined where its denominator equals zero, as division by zero is mathematically invalid.
For \( r(x) = \frac{x^2 - 9}{x^2} \), the denominator is \( x^2 \). Setting this equal to zero, we find that:
  • \( x^2 = 0 \)
  • This simplifies to \( x = 0 \)
Hence, \( x = 0 \) is not in the domain of the function, meaning \( r(x) \) is undefined at this value. This not only affects the search for the y-intercept, confirming its non-existence, but it also signals an important aspect of the graph such as potential vertical asymptotes or points of discontinuity.

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Most popular questions from this chapter

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40] $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}+x^{3}+7 x^{2}+9 x-18 $$

\(59-64=\) A polynomial \(P\) is given. (a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. $$ P(x)=x^{3}-5 x^{2}+4 x-20 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2 $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$
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