Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the domain of the function, (b) decide whether the function is continuous, and (c) identify any horizontal and vertical asymptotes. Verify your answer to part (a) both graphically by using a graphing utility and numerically by creating a table of values. $$f(x)=\frac{4 x^{3}-x^{2}+3}{3 x^{3}+24}$$

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty , +\infty) \), the function is continuous, there is a horizontal asymptote at \( y=\frac{4}{3} \), and there are no vertical asymptotes.

Step 1: Find the Domain

The denominator must not be zero because division by zero is undefined. So solve the equation \(3x^{3}+24=0\). Solving for \(x\), there are no real solutions. Thus, the domain of the function is all real numbers or \( (-\infty , +\infty) \)

Step 2: Check Continuity

As established, the function is defined for all real numbers because there are no values for which the denominator equals zero. Therefore, the function is continuous for its entire domain \( (-\infty , +\infty) \)

Step 3: Identify Horizontal and Vertical Asymptotes

As determined in step 1, there are no vertical asymptotes because our function does not have any points of discontinuity. \nTo find the horizontal asymptotes, analyze the degrees of the polynomial in the numerator and the denominator. In our case, both the numerator and denominator have the same degree (3), so there is a horizontal asymptote at \( y = \frac{a}{b} \), where \( a \) and \( b \) are the coefficients of the highest degree terms in the numerator and the denominator respectively. So, \( y = \frac{4}{3} \) is the horizontal asymptote.

Step 4: Graphical Verification

By plotting the function using a graphing utility, we observe that the function has no breaks or jumps and the graph approaches the line \( y =\frac{4}{3} \) as we move towards positive and negative infinity, confirming the findings.

Step 5: Numerical Verification

Create a table of values that includes numbers within the domain of the function. As \( x \) approaches infinity, both positive and negative, the values of \( f(x) \) will get closer to \( \frac{4}{3} \), reaffirming the presence of a horizontal asymptote at \( y = \frac{4}{3} \)

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented as "x") that the function can accept without running into any issues. For rational functions like our given function \( f(x)=\frac{4x^{3}-x^{2}+3}{3x^{3}+24} \), it's crucial to make sure the denominator is never zero, as dividing by zero is undefined.

In the original exercise, we determined the domain by solving the equation \( 3x^{3}+24=0 \). Since this equation has no real solutions, it means there are no values of \( x \) that would make the denominator zero. Therefore, the domain of this function is all real numbers: \( (-\infty , +\infty) \). This implies that the function can take any real number as input without any restrictions.

• Always check the denominator for zero when dealing with rational functions.
• A domain of all real numbers means there are no restrictions on input values.

Continuity of Functions

Continuity in functions refers to the function's graph being a single, unbroken curve. A function is continuous over an interval if, for every point within that interval, the function does not have any holes, jumps, or asymptotic behaviors.

For the function \( f(x)=\frac{4x^{3}-x^{2}+3}{3x^{3}+24} \), we found that it is continuous over its entire domain \( (-\infty , +\infty) \). Since there are no values that make the denominator zero, there are no breaks or gaps in the function's graph. This means the behavior of the function is predictable across all real numbers.

• A function defined everywhere without breaks or jumps is continuous.
• Check for points where the denominator could potentially be zero to determine discontinuity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches infinity or negative infinity. They indicate the value that the function approaches as you move far along the x-axis in either direction.

For rational functions, the degrees of the polynomials in the numerator and denominator help determine horizontal asymptotes. In our example, both the numerator and denominator polynomials are of degree 3. This means the horizontal asymptote can be found by dividing the leading coefficients. Hence, the horizontal asymptote of \( f(x) \) is \( y = \frac{4}{3} \). This suggests that as \( x \) goes to either positive or negative infinity, the value of \( f(x) \) gets closer and closer to \( \frac{4}{3} \).

• If the degrees of the numerator and denominator are equal, the asymptote is \( y = \frac{a}{b} \), with "a" and "b" as leading coefficients.
• Horizontal asymptotes describe end behavior of functions as \( x \) approaches infinity.

Vertical Asymptotes

Vertical asymptotes occur when a function approaches infinity or negative infinity as the input approaches a certain finite value. This typically happens for rational functions when the denominator of the function goes to zero, resulting in undefined behavior.

For the function \( f(x)=\frac{4x^{3}-x^{2}+3}{3x^{3}+24} \), we determined that there are no vertical asymptotes because the equation \( 3x^{3}+24=0 \) has no real solutions. This implies that \( f(x) \) does not go to infinity at any point within its domain.

• Vertical asymptotes occur at points where the denominator is zero and the numerator is non-zero.
• Since our function's denominator never becomes zero, it has no vertical asymptotes.