Question

Find the domain of the function and identify any horizontal and vertical asymptotes. $$f(x)=\frac{3 x^{2}+1}{x^{2}+9}$$

Short answer

The domain of the function is \((-∞, ∞)\), there are no vertical asymptotes, and the horizontal asymptote is \(y = 3\).

Step-by-step solution

Find the Domain of the Function

The function \(f(x) = \frac{3x^{2} + 1}{x^{2} + 9}\) is defined for all real numbers because the denominator, \(x^{2} + 9\), is never equal to zero for any real number. Therefore, the domain of \(f\) is \(-\infty, \infty\) or in interval notation, \((-∞, ∞)\).

Find the Vertical Asymptotes

A vertical asymptote occurs at the values of \(x\) that make the denominator equal to zero. For the function \(f(x) = \frac{3x^{2} + 1}{x^{2} + 9}\), the denominator \(x^{2} + 9 = 0\) has no real roots. This means that there are no vertical asymptotes.

Find the Horizontal Asymptotes

A horizontal asymptote is determined by the degree of the polynomials. If the degree of the numerator and denominator are the same, as is the case here, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficients for both the numerator and the denominator is \(3\). Therefore, the horizontal asymptote is \(y = \frac{3}{1}\), or \(y = 3\).

Key concepts

Vertical Asymptotes

Vertical asymptotes appear in a function when the denominator approaches zero, which makes the function value shoot off towards infinity or negative infinity. In simpler terms, these are the 'forbidden' points in a function where the output value becomes undefined.

When looking to determine vertical asymptotes, you essentially solve for when the denominator equals zero. Exploring the given example, \(f(x) = \frac{3x^{2} + 1}{x^{2} + 9}\), you will need to find the values of \(x\) that satisfy \(x^{2} + 9 = 0\). However, notice that \(x^{2} + 9\) is always positive since \(x^{2}\) is non-negative; thus it never equals zero.

• No vertical asymptotes exist in this function because the denominator never zeroes out over the set of real numbers.
• In summary, checks for any real values meeting \(x^{2} + 9 = 0\) show none, indicating no vertical asymptote.

Horizontal Asymptotes

Horizontal asymptotes occur when you evaluate the behavior of a function as \(x\) approaches infinity or negative infinity. In practice, these are the lines that the graph of a function gets closer to, a kind of leveling out, as more extreme values of \(x\) go into play.

For rational functions, such as \(f(x) = \frac{3x^{2} + 1}{x^{2} + 9}\), horizontal asymptotes depend on comparing the degree (highest power term) of the numerator and denominator.

• If the degree of the numerator is less than the denominator, the asymptote is \(y = 0\).
• If the degree of the numerator equals the denominator, like here where both are 2, the horizontal asymptote is confirmed through the coefficients: \(y = \frac{3}{1} = 3\).
• If the numerator's degree exceeds the denominator's, no horizontal asymptote is present. Instead, oblique asymptotes could occur.

For our function, the horizontal asymptote is \(y = 3\). The graph of \(f(x)\) will extend ever closer to this line as \(x\) becomes increasingly large, demonstrating a consistent boundary edge to this polynomial function's graph.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number exponents multiplied by coefficients, combined in sum or difference. Simply put, they are a series of "terms" combined, each term containing variable powers with coefficients.

The basic form of a polynomial in a single variable could look something like \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). The largest power of \(x\) seen in a function is termed the degree of that polynomial.

• Example: In the polynomial \(3x^{2} + 1\), the degree is 2.
• Polynomial degrees dictate the behavior and type of asymptote a rational function might exhibit.

Rational functions often stem from polynomials like in \(f(x) = \frac{3x^{2} + 1}{x^{2} + 9}\), where both numerator and denominator are polynomial expressions. The balance of powers (degrees) between these dictates the existence of horizontal asymptotes.
Understanding polynomial structure aids in predicting function behavior, and especially in finding asymptotes which act as critical guideposts in graph analysis.