Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4-3x)/(2x+1)

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Identify the rational function given: \(f(x) = \frac{4 - 3x}{2x + 1}\).

Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): \(2x + 1 = 0\). Solve this equation to find the value(s) of \(x\) where the function is undefined.

Determine the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. The numerator degree is the highest power of \(x\) in \(4 - 3x\), and the denominator degree is the highest power of \(x\) in \(2x + 1\).

Since the degrees of numerator and denominator are equal (both degree 1), find the horizontal asymptote by dividing the leading coefficients of the numerator and denominator. The horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).

If the degree of the numerator were exactly one more than the degree of the denominator, you would perform polynomial long division to find the oblique asymptote. In this case, since degrees are equal, no oblique asymptote exists.

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Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions, especially where the denominator is zero, is essential for identifying asymptotes and graphing the function.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. These lines are vertical and indicate values of x that the function cannot take.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the end behavior of a function as x approaches infinity or negative infinity, determined by comparing degrees of numerator and denominator. Oblique (slant) asymptotes occur when the numerator's degree is exactly one more than the denominator's, found via polynomial division.