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Chapter 5: Problem 24

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{94-2 x^{2}}{3 x^{2}-12} $$

Short Answer

Expert verified
x-intercepts: \(x=\sqrt{47}, -\sqrt{47}\); y-intercept: \(y=-\frac{47}{6}\).

Step by step solution

01

Find the x-intercepts

To find the x-intercepts of the function, set the numerator equal to zero and solve for \( x \). The function is \( f(x)=\frac{94-2x^2}{3x^2-12} \). Thus, solve \( 94-2x^2=0 \).Rearrange to \( 2x^2=94 \).Divide by 2: \( x^2 = 47 \).Taking the square root gives \( x = \pm \sqrt{47} \). Hence, the x-intercepts are \( x = \sqrt{47} \) and \( x = -\sqrt{47} \).
02

Simplify To Find the y-intercept

To find the y-intercept, evaluate \( f(x) \) at \( x = 0 \). Substitute \( x = 0 \) into the function:\( f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12} = \frac{94}{-12} \).Simplify the fraction: \( f(0) = \frac{47}{-6} = -\frac{47}{6} \).Thus, the y-intercept is \( y = -\frac{47}{6} \).
03

Finalize Intercepts

To summarize, the coordinates of the x-intercepts are \( (\sqrt{47}, 0) \) and \( (-\sqrt{47}, 0) \). The y-intercept is at \( (0, -\frac{47}{6}) \). These points can be used to sketch the graph and understand where the function crosses the axes.

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Key Concepts

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

x-intercepts
The x-intercepts of a function are points where the graph crosses or touches the x-axis. This occurs when the value of the function is zero.
To find the x-intercepts of an algebraic or rational function, you need to set the numerator equal to zero and solve for the variable, usually represented by \( x \).
For the function given in the example, \( f(x)=\frac{94-2x^2}{3x^2-12} \), this process involved the following steps:
  • Set the numerator, \( 94 - 2x^2 \), equal to zero: \( 94 - 2x^2 = 0 \).
  • This leads to the equation \( 2x^2 = 94 \).
  • By dividing both sides by 2, we get \( x^2 = 47 \).
  • Taking the square root of both sides, we find \( x = \pm \sqrt{47} \).
Thus, the x-intercepts are \( x = \sqrt{47} \) and \( x = -\sqrt{47} \). These represent the points \( (\sqrt{47}, 0) \) and \( (-\sqrt{47}, 0) \) on the coordinate plane.
y-intercepts
Y-intercepts occur where the graph crosses the y-axis. At this point, the value of \( x \) is always zero, thus to find the y-intercept of any function, simply substitute \( x = 0 \) into the function and solve for \( y \).
In rational functions like our example \( f(x)=\frac{94-2x^2}{3x^2-12} \), this is straightforward:
  • Substitute \( x = 0 \) into the function, giving \( f(0) = \frac{94-2(0)^2}{3(0)^2-12} \).
  • Simplify this expression to \( f(0) = \frac{94}{-12} \).
  • Reducing the fraction results in \( f(0) = -\frac{47}{6} \).
Thus, the y-intercept is \( y = -\frac{47}{6} \), represented by the point \( (0, -\frac{47}{6}) \), showing where the graph intersects the y-axis.
rational functions
Rational functions are ratios of two polynomials. They take the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).
These functions can behave differently at their intercepts, vertical asymptotes, and horizontal asymptotes.
  • The intercepts, as discussed previously, occur where the graph crosses the axes.
  • Vertical asymptotes occur where the denominator \( q(x) \) equals zero, provided it doesn't simultaneously zero the numerator.
  • Horizontal asymptotes depend on the degrees of \( p(x) \) and \( q(x) \).
    • If the degree of \( p(x) \) is less than that of \( q(x) \), the x-axis \( y = 0 \) is the asymptote.
    • If the degree of \( p(x) \) equals that of \( q(x) \), the horizontal asymptote is at \( y = \frac{text{leading coefficient of } p(x)}{text{leading coefficient of } q(x)} \).
    • Finally, if the degree of \( p(x) \) is greater than that of \( q(x) \), there is no horizontal asymptote, but possibly a slant asymptote.
This foundational understanding of rational functions is essential for analyzing and graphing these functions accurately.

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Most popular questions from this chapter

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the square root of \(x .\) When \(x=64,\) then \(y=12 .\) Find \(y\) when \(x=36\).

For the following exercises, find the inverse of the functions. $$ f(x)=x^{2}+2 x,[-1, \infty) $$

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as the square of \(x\) and when \(x=3, \quad y=2\).

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly with the square of \(x\) and when \(x=2, y=3\).

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square root of \(x\) and when \(x=36, \quad y=24\).
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