Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{x+5}{x^{2}+4} $$

Step-by-Step Solution

Answer

The x-intercept is \((-5, 0)\) and the y-intercept is \(\left(0, \frac{5}{4}\right)\).

Step 1: Find the x-intercept

To find the x-intercept of the function, set the numerator of the function equal to zero and solve for \(x\). The function is \(f(x) = \frac{x+5}{x^2 + 4}\). Set \(x+5 = 0\) and solve for \(x\):\[x+5=0\]This gives us \(x = -5\). So, the x-intercept is at the point \((-5, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate the function at \(x = 0\). Substitute \(x = 0\) into the function:\[f(0) = \frac{0+5}{0^2+4} = \frac{5}{4}\]So, the y-intercept is at the point \(\left(0, \frac{5}{4}\right)\).

Key Concepts

Finding x-intercepts

To find the x-intercepts of a function, our main goal is to determine where the graph of the function crosses the x-axis. These points occur when the output of the function, usually denoted as \(f(x)\), equals zero. In simpler terms, we're looking for inputs (x-values) that make the numerator of the function zero, while ensuring the denominator is not zero at these points because division by zero is undefined.

Here's the step-by-step approach to finding x-intercepts:

• Set the numerator equal to zero: For the function \(f(x) = \frac{x+5}{x^2+4}\), set \(x+5 = 0\).
• Solve for x: Solving \(x+5=0\) gives \(x = -5\).
• Check the denominator: Ensure that this x-value does not make the denominator zero. Here, \(x^2+4\) is never zero for any real number \(x\), as squares of real numbers are non-negative and adding 4 keeps it positive.

Thus, the x-intercept is at the point \((-5, 0)\). It’s a great way to understand how the roots of the numerator impact the function’s graph. By solving the numerator, we learn where to expect the function to touch the x-axis.

Finding y-intercepts

Y-intercepts are the points where a function's graph crosses the y-axis. This occurs when the input, or x-value, is zero. Finding the y-intercept of a function is typically straightforward and involves simply evaluating the function when \(x=0\).

Here’s a concise guide on how to find y-intercepts:

• Substitute x with zero: In our function \(f(x) = \frac{x+5}{x^2+4}\), substitute \(x=0\).


• Solve the expression: Calculate \(f(0) = \frac{0+5}{0^2+4} = \frac{5}{4}\).

This means the y-intercept of the function is at the point \(\left(0, \frac{5}{4}\right)\). It's a vital concept because it provides a specific starting point or reference when examining or graphing the behavior of the function. Whenever you graph a function, locating the y-intercept helps ensure accuracy in illustrating where the graph begins along the y-axis.

Rational functions

Rational functions are a fascinating topic in algebra, widely used in various mathematical models. A rational function is expressed as the ratio of two polynomials. Such functions take the form \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero.

Understanding rational functions involves recognizing their properties:

• Domain: Every rational function must have a denominator that isn’t zero. This defines the domain as all real numbers except where \(Q(x) = 0\).
• X- and y-intercepts: As explained in earlier sections, we find these intercepts by setting the numerator to zero for x-intercepts, and substituting zero into the function for y-intercepts.
• Asymptotes: Rational functions often have vertical asymptotes defined by the zeros of the denominator, and sometimes horizontal or slant asymptotes which depend on the relationship of the degrees of \(P(x)\) and \(Q(x)\).

Rational functions are versatile and appear frequently in real-life scenarios, like rate and proportion problems. Understanding these functions helps in interpreting behaviors of numerous phenomena. By examining their intercepts and asymptotes, you grasp both the basic layout and detailed features of their graphs.