To identify the graph of the function [tex]\( f(x) = \frac{10 - 10x^2}{x^2} \)[/tex], we'll proceed through several steps: setting simpler forms, identifying key characteristics, and sketching the behavior. Let's break this down step by step.
### Step 1: Simplify the Function
First, simplify the function to help understand its behavior.
[tex]\[ f(x) = \frac{10 - 10x^2}{x^2} \][/tex]
We can split the expression into two separate fractions:
[tex]\[ f(x) = \frac{10}{x^2} - \frac{10x^2}{x^2} \][/tex]
Simplify further:
[tex]\[ f(x) = \frac{10}{x^2} - 10 \][/tex]
### Step 2: Determine the Domain
Identify the domain of the function. Since [tex]\( x^2 \)[/tex] is in the denominator, [tex]\( x \)[/tex] cannot be zero:
[tex]\[ x \neq 0 \][/tex]
So the domain is all real numbers except zero, or [tex]\( (-\infty, 0) \cup (0, +\infty) \)[/tex].
### Step 3: Analyze Asymptotes
1. Vertical Asymptote:
The function is undefined at [tex]\( x = 0 \)[/tex]. Thus, there is a vertical asymptote at [tex]\( x = 0 \)[/tex].
2. Horizontal Asymptote:
To see if there is a horizontal asymptote, consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \to \pm \infty \)[/tex]:
[tex]\[ \lim_{x \to \pm \infty} f(x) = \lim_{x \to \pm \infty} \left(\frac{10}{x^2} - 10\right) \][/tex]
As [tex]\( x \to \pm \infty \)[/tex], [tex]\( \frac{10}{x^2} \to 0 \)[/tex]:
[tex]\[ \lim_{x \to \pm \infty} f(x) = -10 \][/tex]
Thus, there is a horizontal asymptote at [tex]\( y = -10 \)[/tex].
### Step 4: Analyze Intercepts
1. x-intercepts:
For [tex]\( f(x) \)[/tex] to have x-intercepts, solve [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \frac{10}{x^2} - 10 = 0 \][/tex]
[tex]\[ \frac{10}{x^2} = 10 \][/tex]
[tex]\[ 10 = 10x^2 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
[tex]\[ x = \pm 1 \][/tex]
Therefore, the x-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (1, 0) \)[/tex].
2. y-intercepts:
There is no y-intercept because [tex]\( f(x) \)[/tex] is not defined at [tex]\( x = 0 \)[/tex].
### Step 5: Determine the Behavior of [tex]\( f(x) \)[/tex] Close to Vertical Asymptote
As [tex]\( x \to 0^+ \)[/tex] or [tex]\( x \to 0^- \)[/tex], we need to see how the function behaves:
[tex]\[ f(x) = \frac{10}{x^2} - 10 \][/tex]
- As [tex]\( x \to 0^+ \)[/tex] and [tex]\( x \to 0^- \)[/tex], [tex]\( \frac{10}{x^2} \)[/tex] becomes very large.
- Thus, [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
### Step 6: Sketch the Graph
1. Plot vertical asymptote at [tex]\( x = 0 \)[/tex].
2. Identify horizontal asymptote at [tex]\( y = -10 \)[/tex].
3. Locate the x-intercepts at [tex]\( (-1, 0) \)[/tex] and [tex]\( ( 1, 0) \)[/tex].
4. Plot the general behavior:
- As [tex]\( x \)[/tex] approaches 0 from the right and left, [tex]\( f(x) \)[/tex] shoots towards [tex]\( +\infty \)[/tex].
- As [tex]\( x \)[/tex] moves away from the vertical asymptote towards [tex]\(\pm \infty \)[/tex], the function approaches the horizontal asymptote [tex]\( y = -10 \)[/tex].
### Conclusion
Putting all this information together, we can sketch the graph of [tex]\( f(x) \)[/tex]:
- The function will show a steep increase as it approaches [tex]\( x=0 \)[/tex] from either side.
- It will cross the x-axis at [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
- Moving away from the center, the curve will asymptotically approach [tex]\( y = -10 \)[/tex].
This is a rational function with characteristics typical of hyperbolic curves with an infinite behavior at the vertical asymptote and flattening out towards the horizontal asymptote.
