Problem 478 - Precalculus

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$f(x)=\frac{94-2 x^{2}}{3 x^{2}-12}$$

Step-by-Step Solution

Answer

X-intercepts: \(x=\pm\sqrt{47}\). Y-intercept: \(y=-\frac{47}{6}\).

Step 1: Find the x-intercepts

The x-intercepts of a function are the points where the function crosses the x-axis. This occurs where the output of the function is zero: that is, solve for \( x \) when \( f(x) = 0 \). Set the numerator equal to zero because a fraction is zero when its numerator is zero: \[ 94 - 2x^2 = 0. \]Solve for \( x \):1. Add \( 2x^2 \) to both sides: \[ 94 = 2x^2. \]2. Divide both sides by 2: \[ 47 = x^2. \]3. Take the square root of both sides: \[ x = \pm \sqrt{47}. \]Therefore, the x-intercepts are at \( x = \sqrt{47} \) and \( x = -\sqrt{47} \).

Step 2: Find the y-intercepts

The y-intercepts of a function are the points where the function crosses the y-axis. This occurs when \( x = 0 \). Substitute \( x = 0 \) into the function \( f(x) \):\[ f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12} = \frac{94}{-12} = -\frac{47}{6}. \]Thus, the y-intercept is \( y = -\frac{47}{6} \).

Step 3: State the Intercepts

Now that we've calculated both intercepts, summarize the results. The x-intercepts are \( x = \sqrt{47} \) and \( x = -\sqrt{47} \). The y-intercept is \( y = -\frac{47}{6} \).

Key Concepts

Understanding the x-intercepts

The x-intercepts of a function are where the graph crosses the x-axis. At these points, the function equals zero. Think of the x-intercept as the spot where you can imagine zooming down to zero on the y-axis while still being somewhere along the x-axis.

To find them, you set the entire function equal to zero. For the given exercise function \(f(x)=\frac{94-2x^{2}}{3x^{2}-12}\), the rule is simple: the fraction equals zero when the numerator is zero!
So, solve \(94 - 2x^2 = 0\). This makes things a bit easier, doesn't it?

• Add \(2x^2\) to 94, making it \(94 = 2x^2\).
• Then, divide by 2: \(47 = x^2\).
• Taking the square root gives \(x = \pm \sqrt{47}\).

This means the x-intercepts are at \(x = \sqrt{47}\) and \(x = -\sqrt{47}\). Each intercept indicates a point where the function bounces off or cuts through the x-axis.

Exploring the y-intercepts

The y-intercepts happen where the graph crosses the y-axis. At this point, the x-value is always zero since it's the point directly up or down from the origin along the y-axis.

Finding a y-intercept is straightforward: simply set \(x = 0\) in the function and solve. Using our exercise's function \(f(x)=\frac{94-2x^{2}}{3x^{2}-12}\), replace every instance of \(x\) with zero:

• This gives \(f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12}\).
• Simplify to \(f(0) = \frac{94}{-12}\).
• Upon simplifying further, you have \(y = -\frac{47}{6}\).

This simple substitution shows the y-intercept as \(y = -\frac{47}{6}\). It indicates where the function crosses the y-axis, providing us with crucial plot points.

Equations and solving them easily

Solving equations is like solving puzzles. You follow clues that help you find what makes the equation true or equal. In our example, we worked with a function in the form of a rational expression, which involves solving both numerators and denominators strategically.

Let's look at solving an equation in practical terms:

• Focus first on simplifying each side of the equation.
• Move terms around to isolate variables when possible.
• Use operations such as addition, subtraction, multiplication, and division to get a variable by itself.
• Sometimes, you'll take additional steps like factoring or using the square root.

In the exercise solution:

• For the x-intercepts, solving \(94 - 2x^2 = 0\) followed these steps: gathering like terms, dividing to simplify, and taking a square root to find \(x\) values.
• That's quite handy when handling quadratic forms.

In general, these strategies help reliably solve similar exercises and understand functional intercepts better.