Select ALL The Information That Applies To The Given Rational Function.$\[ F(x)=\frac{2x^2-2}{x^2-4x+3} \\]Select All Correct Options:- \[$x\$\]-intercepts At \[$(3, 0)\$\]- Hole At \[$(1, 0)\$\]- Vertical Asymptote At

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Understanding Rational Functions

A rational function is a type of function that is defined as the ratio of two polynomials. It is represented in the form of f(x)=p(x)q(x){ f(x) = \frac{p(x)}{q(x)} }, where p(x){ p(x) } and q(x){ q(x) } are polynomials. Rational functions can have various characteristics, such as intercepts, holes, and asymptotes, which are essential to understand when analyzing these functions.

Given Rational Function

The given rational function is f(x)=2x2−2x2−4x+3{ f(x) = \frac{2x^2 - 2}{x^2 - 4x + 3} }. To analyze this function, we need to identify its characteristics, such as intercepts, holes, and asymptotes.

Intercepts

Intercepts are the points where the graph of the function intersects the x-axis or y-axis. To find the x-intercepts, we need to set the numerator of the function equal to zero and solve for x.

Finding x-Intercepts

To find the x-intercepts, we set the numerator 2x2−2{ 2x^2 - 2 } equal to zero and solve for x.

2x2−2=0{ 2x^2 - 2 = 0 }

2x2=2{ 2x^2 = 2 }

x2=1{ x^2 = 1 }

x=±1{ x = \pm 1 }

Therefore, the x-intercepts of the function are at (−1,0){ (-1, 0) } and (1,0){ (1, 0) }.

Finding y-Intercepts

To find the y-intercepts, we need to evaluate the function at x = 0.

f(0)=2(0)2−2(0)2−4(0)+3{ f(0) = \frac{2(0)^2 - 2}{(0)^2 - 4(0) + 3} }

f(0)=−23{ f(0) = \frac{-2}{3} }

Therefore, the y-intercept of the function is at (0,−23){ (0, -\frac{2}{3}) }.

Holes

Holes are the points where the function is not defined, but the graph passes through that point. To find the holes, we need to factor the denominator and identify the points where the function is not defined.

Factoring the Denominator

The denominator of the function is x2−4x+3{ x^2 - 4x + 3 }. We can factor this expression as:

x2−4x+3=(x−1)(x−3){ x^2 - 4x + 3 = (x - 1)(x - 3) }

Identifying Holes

From the factored form of the denominator, we can see that the function is not defined when x=1{ x = 1 } or x=3{ x = 3 }. However, since the numerator is not equal to zero at these points, there are no holes at x=1{ x = 1 } or x=3{ x = 3 }.

Vertical Asymptotes

Vertical asymptotes are the vertical lines that the graph of the function approaches as x approaches a certain value. To find the vertical asymptotes, we need to identify the points where the denominator is equal to zero.

Finding Vertical Asymptotes

From the factored form of the denominator, we can see that the denominator is equal to zero when x=1{ x = 1 } or x=3{ x = 3 }. Therefore, the vertical asymptotes of the function are at x=1{ x = 1 } and x=3{ x = 3 }.

Conclusion

In conclusion, the given rational function f(x)=2x2−2x2−4x+3{ f(x) = \frac{2x^2 - 2}{x^2 - 4x + 3} } has x-intercepts at (−1,0){ (-1, 0) } and (1,0){ (1, 0) }, but not at (3,0){ (3, 0) }. There are no holes at (1,0){ (1, 0) } or (3,0){ (3, 0) }, but there are vertical asymptotes at x=1{ x = 1 } and x=3{ x = 3 }.

Selecting Correct Options

Based on the analysis of the given rational function, we can select the correct options as follows:

  • x-intercepts at (−1,0){ (-1, 0) } and (1,0){ (1, 0) }
  • No hole at (1,0){ (1, 0) }
  • Vertical asymptote at x=1{ x = 1 }
  • No vertical asymptote at x=3{ x = 3 }

Therefore, the correct options are:

  • x-intercepts at (−1,0){ (-1, 0) } and (1,0){ (1, 0) }
  • No hole at (1,0){ (1, 0) }
  • Vertical asymptote at x=1{ x = 1 }
    Rational Function Analysis: Q&A =====================================

Q: What is a rational function?

A: A rational function is a type of function that is defined as the ratio of two polynomials. It is represented in the form of f(x)=p(x)q(x){ f(x) = \frac{p(x)}{q(x)} }, where p(x){ p(x) } and q(x){ q(x) } are polynomials.

Q: What are the characteristics of a rational function?

A: Rational functions can have various characteristics, such as intercepts, holes, and asymptotes. Intercepts are the points where the graph of the function intersects the x-axis or y-axis. Holes are the points where the function is not defined, but the graph passes through that point. Asymptotes are the vertical lines that the graph of the function approaches as x approaches a certain value.

Q: How do you find the x-intercepts of a rational function?

A: To find the x-intercepts, we need to set the numerator of the function equal to zero and solve for x.

Q: How do you find the y-intercepts of a rational function?

A: To find the y-intercepts, we need to evaluate the function at x = 0.

Q: What is a hole in a rational function?

A: A hole is a point where the function is not defined, but the graph passes through that point.

Q: How do you find the holes in a rational function?

A: To find the holes, we need to factor the denominator and identify the points where the function is not defined.

Q: What is a vertical asymptote in a rational function?

A: A vertical asymptote is a vertical line that the graph of the function approaches as x approaches a certain value.

Q: How do you find the vertical asymptotes of a rational function?

A: To find the vertical asymptotes, we need to identify the points where the denominator is equal to zero.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote.

Q: Can a rational function have a hole and a vertical asymptote at the same point?

A: No, a rational function cannot have a hole and a vertical asymptote at the same point.

Q: How do you graph a rational function?

A: To graph a rational function, we need to identify the x-intercepts, y-intercepts, holes, and vertical asymptotes. We can then use this information to sketch the graph of the function.

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

  • Not factoring the denominator correctly
  • Not identifying the holes and vertical asymptotes correctly
  • Not evaluating the function at the correct points
  • Not using the correct notation for the function

Q: How do you simplify a rational function?

A: To simplify a rational function, we need to factor the numerator and denominator and cancel out any common factors.

Q: Can a rational function be simplified to a polynomial?

A: Yes, a rational function can be simplified to a polynomial if the denominator is a constant.

Q: Can a rational function be simplified to a linear function?

A: Yes, a rational function can be simplified to a linear function if the numerator and denominator are both linear.