Problem 474 - Precalculus

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 (0, 5/4).

Step 1: Identify the x-intercept

To find the x-intercept, set the function equal to zero and solve for \(x\). Thus, we set \(f(x) = \frac{x+5}{x^{2}+4} = 0\). The fraction is zero when the numerator is zero (assuming the denominator is not zero). Therefore, solve for \(x + 5 = 0\).

Step 2: Solve for x-intercept

Solve the equation \(x + 5 = 0\). Subtract 5 from both sides to get \(x = -5\). Hence, the x-intercept is at \((-5, 0)\).

Step 3: Identify the y-intercept

To find the y-intercept, evaluate the function at \(x = 0\). Substitute \(x = 0\) in the function: \(f(0) = \frac{0 + 5}{0^{2} + 4} = \frac{5}{4}\).

Step 4: State the y-intercept

The y-intercept is the value of the function when \(x=0\). Hence, the y-intercept is at \((0, \frac{5}{4})\).

Key Concepts

Understanding X-Intercepts

X-intercepts are the points where the graph of a function crosses the x-axis. This happens when the output of the function, or the y-value, is zero. In simpler terms, an x-intercept occurs where the function equals zero.

• To find the x-intercept, you must set the entire function equal to zero and solve for x.
• For example, consider the function given: \[ f(x) = \frac{x+5}{x^{2}+4} \]
• You set this equation to zero to find the x-value where the y-value is zero:
\[ \frac{x+5}{x^{2}+4} = 0 \]
• Focus on the numerator \(x + 5 = 0\) because a fraction can only be equal to zero when the numerator equals zero and the denominator is not zero.
• Solving \(x + 5 = 0\) by subtracting 5 from each side, we find that \(x = -5\).

Therefore, the x-intercept of the function is at (-5, 0). This means if you were to plot the function on a graph, it would cross the x-axis at this point.

Understanding Y-Intercepts

Y-intercepts are points where the graph of a function crosses the y-axis. This occurs when the x-value is zero. Typically, finding the y-intercept involves evaluating the function when x equals zero.

• To find the y-intercept, simply plug x = 0 into the function.
• In the context of the example function, we evaluate:
\[ f(0) = \frac{0 + 5}{0^{2} + 4} \]
• This simplifies to \( \frac{5}{4} \).

Thus, the y-intercept for the function is at the point (0, \( \frac{5}{4} \)). On a graph, this means that the function intersects the y-axis at this point. It's important to understand that y-intercepts provide a snapshot of the function's value when no input is applied (or when x is zero), offering a starting point to visualize the function's behaviour.

Introduction to Rational Functions

Rational functions are a type of function formed by the ratio of two polynomials. They can often be identified by having the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. Understanding the composition and behavior of rational functions is essential for finding intercepts and analyzing the function's graph.

• These functions can have asymptotes, which are lines the graph approaches but never touches, due to the nature of the denominator.
• Analyzing rational functions involves checking for both vertical and horizontal asymptotes.
• X-intercepts are found where the numerator equals zero (assuming no common factors cancel out with the denominator).
• Y-intercepts are determined by setting x to zero in the entire function.

For example, in the function \( f(x) = \frac{x+5}{x^{2}+4} \):

• It is already in the form of a rational function, where \(P(x) = x+5\) and \(Q(x)=x^{2}+4\).
• It helps to first identify potential asymptotes by analyzing where \(Q(x)\) might equal zero.

Rational functions offer rich mathematical insights and are vital for higher-level mathematics and real-world applications, such as modeling rates and proportions.