Complete the following steps for the given functions.

b. Find the vertical asymptotes of f (if any).

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

검증된 단계별 안내

Identify the vertical asymptotes by finding the values of x that make the denominator zero.

Set the denominator equal to zero: 3x + 4 = 0.

Solve the equation for x: 3x = -4.

Divide both sides by 3 to isolate x: x = -$\frac{4}{3}$.

Conclude that the vertical asymptote is at x = -$\frac{4}{3}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator does not. These points indicate where the function's value tends to infinity or negative infinity. To find vertical asymptotes, set the denominator equal to zero and solve for the variable, ensuring that the numerator does not also equal zero at those points.

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions, particularly their asymptotic behavior, is crucial for analyzing their graphs and identifying points of discontinuity.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. The degree of a polynomial indicates its highest power, which influences its end behavior and the number of roots it can have. In the context of rational functions, the behavior of the numerator polynomial is essential for determining limits and asymptotic behavior.

Complete the following steps for the given functions. b. Find the vertical asymptotes of f (if any).f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)