Problem 20 - College Algebra

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

x-intercept: (-5, 0), y-intercept: (0, 5/4).

Step 1: Find the y-intercept

The y-intercept of a function is found by evaluating the function at \(x = 0\). Substitute \(x = 0\) into the function: \(f(0) = \frac{0 + 5}{0^2 + 4} = \frac{5}{4}\). Thus, the y-intercept is \(\left(0, \frac{5}{4}\right)\).

Step 2: Find the x-intercept

The x-intercept of a function is found where \(f(x) = 0\). Set the numerator equal to zero and solve for \(x\): \(x + 5 = 0\). This gives \(x = -5\). Therefore, the x-intercept is \((-5, 0)\).

Key Concepts

Understanding the x-intercept

The x-intercept of a function is a point where the graph of the function crosses the x-axis. At this point, the value of the function, represented as \(f(x)\), is zero. In simpler terms, it’s the point where the output of the function equals zero, making the coordinate \((x, 0)\). Let’s look at the function \(f(x)=\frac{x+5}{x^{2}+4}\) to understand how to find the x-intercept.

To find the x-intercept, we need the function to equal zero. For rational functions like this, \(\frac{x+5}{x^2+4} = 0\), the output is zero when the numerator is zero while the denominator is not zero. Thus, set the numerator \(x + 5\) equal to zero and solve for \(x\). This gives us the equation \(x + 5 = 0\). Solving this equation, we find that \(x = -5\). So, the x-intercept is at the point \((-5, 0)\).

Remember, the denominator \(x^2+4\) does not affect the x-intercept because it should not be zero (as that would make the function undefined). The sole focus is the numerator in this context.

Understanding the y-intercept

The y-intercept is the point where a graph of a function intercepts the y-axis. At this point, the value of \(x\) is zero. In other words, it represents what value the function \(f(x)\) takes when \(x = 0\). For any function, finding the y-intercept can be as straightforward as substituting zero in place of \(x\) in the function’s equation.

Let's apply this to the function \(f(x)=\frac{x+5}{x^{2}+4}\). By substituting \(x = 0\), we have:

• \(f(0) = \frac{0 + 5}{0^2 + 4}\)
• This simplifies to \(f(0) = \frac{5}{4}\)

Thus, the y-intercept of the function is \((0, \frac{5}{4})\).

Being able to find the y-intercept quickly is useful for sketching graphs, as it provides a starting touchpoint on the graph of a function.

Exploring Rational Functions

Rational functions are a type of function represented by the ratio of two polynomials. Essentially, if you have a function \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, you are dealing with a rational function. These functions can exhibit interesting properties including asymptotes, intercepts, and domain restrictions.

A key aspect of handling rational functions is understanding their intercepts, as explored with \(f(x)=\frac{x+5}{x^{2}+4}\). Since the denominator \(x^2 + 4\) never equals zero for real numbers \(x\) (because \(x^2 + 4\) is always positive), this function is defined for all real numbers. This makes the calculation of intercepts straightforward because you aren’t concerned with undefined points due to division by zero.

When working with rational functions:

• The numerator determines the x-intercepts.
• The entire expression, including how it relates to zero, determines the y-intercept by evaluating it at \(x=0\).

Grasping these basics helps in sketching the graph, predicting behavior, and solving practical problems involving rational functions.