Exploring Rational Functions

What is the definition of a rational function?

• A rational function is a function of the form f(x) = P(x)/Q(x), where P and Q are polynomials and Q(x) ≠ 0.

• A rational function is a function defined only for integer values of x.

• A rational function is a function that has no variables in the denominator.

• A rational function is a function that can be expressed as a sum of polynomials.

How do you identify the vertical asymptotes of a rational function?

• Set the numerator of the rational function to zero and solve for the variable.

• Set the denominator of the rational function to zero and solve for the variable.

• Identify the maximum value of the function and use it to find asymptotes.

• Graph the function and visually estimate where the asymptotes are located.

What is the horizontal asymptote of the function f(x) = (2x^2 + 3)/(x^2 - 1)?

• y = 3

• y = 1

• y = 0

• y = 2

Explain how to find the x-intercepts of a rational function.

• Set the denominator of the rational function to zero and solve for x.

• Find the vertex of the function and use it to determine the x-intercepts.

• Graph the function and read the x-intercepts directly from the graph.

• Set the numerator of the rational function to zero and solve for x.

What is the domain of the function f(x) = 1/(x - 4)?

• All real numbers

• Only positive numbers

• All real numbers except 0

• All real numbers except 4

How do you determine if a rational function is even, odd, or neither?

• Find the roots of the function.

• Graph the function and observe its symmetry.

• Check the degree of the numerator and denominator.

• Evaluate f(-x) and compare with f(x) and -f(x) to classify the function.

What is the significance of the numerator and denominator in a rational function?

• The numerator determines the output, and the denominator indicates where the function is undefined.

• The numerator shows the function's domain, and the denominator shows the range.

• The numerator indicates the function's symmetry, while the denominator shows its continuity.

• The numerator and denominator both determine the function's maximum value.

How can you simplify the rational expression (x^2 - 9)/(x + 3)?

• x^2 + 9

• x - 3

• x + 3

• x - 9

What is the range of the function f(x) = (3x)/(x^2 + 1)?

• (-1, 1)

• (-2, 2)

• (0, 3)

• (-3/2, 3/2)

Describe how to graph a rational function step by step.

• Use only the first derivative to determine the graph's shape.

• Only plot the function's maximum and minimum points.

• Ignore asymptotes and focus solely on the intercepts.

• Graphing a rational function involves identifying the function's domain, intercepts, asymptotes, and plotting key points.