Problem 15 - Algebra and Trigonometry

Question

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{x^{2}-9}{x^{2}} $$

Step-by-Step Solution

Answer

The x-intercepts are (3,0) and (-3,0), and there is no y-intercept.

Step 1: Understanding Intercepts

The x-intercept of a function occurs where the function crosses the x-axis. This means the output, or y-value, is zero. To find it for a rational function, set the numerator equal to zero: \(x^2 - 9 = 0\). The y-intercept occurs where the function crosses the y-axis, which is where the input, or x-value, is zero: substitute \(x = 0\) in the function.

Step 2: Find x-intercepts

Set the numerator \(x^2 - 9\) equal to zero and solve for \(x\): \[ x^2 - 9 = 0 \] This can be rewritten as \((x+3)(x-3) = 0\). Solving this gives \(x = 3\) and \(x = -3\), thus the x-intercepts are at \((3,0) \) and \((-3,0)\).

Step 3: Find y-intercept

To find the y-intercept, substitute \(x = 0\) into the function:\[ r(0) = \frac{0^2 - 9}{0^2} \] This expression is undefined because the denominator is zero. Since division by zero is not possible, there is no y-intercept for this function.

Key Concepts

x-intercepts

When dealing with rational functions, identifying the x-intercepts involves a key step: finding out where the function touches or crosses the x-axis. These points occur when the output, or y-value, of the function is zero. For rational functions, the x-intercepts can be found by setting the numerator equal to zero, as it directly controls the output. This is because any number divided by zero results in zero.
For the function given, \( r(x) = \frac{x^2 - 9}{x^2} \), we set the numerator equal to zero:

• \( x^2 - 9 = 0 \).

The equation \( x^2 - 9 = 0 \) can be factored as \((x+3)(x-3) = 0\). Solving this, we find the x-values that satisfy the equation are 3 and -3. Thus, the x-intercepts of the function are at the points (3, 0) and (-3, 0). This means the graph of the function crosses the x-axis at these points.

y-intercepts

Finding y-intercepts for a rational function is slightly different from finding x-intercepts. The y-intercept is the point where the graph of the function crosses the y-axis. This special point occurs when the input, or x-value, is zero. To determine if there is a y-intercept, you substitute zero for x in the function.
For the given function \( r(x) = \frac{x^2 - 9}{x^2} \), substituting \( x = 0 \) results in:

• \( r(0) = \frac{0^2 - 9}{0^2} \)

Unfortunately, this expression is undefined due to the presence of zero in the denominator. In mathematical terms, division by zero is undefined, which means calculating this is impossible. Hence, our function does not have a y-intercept, as the point at which x is zero does not exist within the scope of the function.

solving equations

Solving equations is a fundamental skill, and in the context of rational functions, it involves setting and solving specific parts of the function to find particular values. To find x-intercepts, as discussed earlier, you solve for when the numerator equals zero, since that results in the function's output being zero.
In this example, the equation \( x^2 - 9 = 0 \) was solved:

• Rewritten as \((x+3)(x-3) = 0\).

Solving the factored form results in x-values of 3 and -3. This directly gives the x-intercepts.
The scenario regarding y-intercepts involves substituting x = 0 into the function, although this led to division by zero (which teaches us about cases where something cannot be solved straightforwardly due to restrictions on certain expressions like division by zero). It's crucial to remember these problem-solving processes can sometimes highlight limits and behaviors in functions, allowing us to fully comprehend the nature of rational equations and their graphs.