### Analyzing and Understanding Rational Functions: A Case Study on \( f(x) \)#### Problem Statement:Given the equation for the following rational function, select **all** properties that are true for the graph of \( f(x) \).\[ f(x) = \frac{x^2 + 4x - 12}{3x^2 - 12} \]Through the list of statements below, identify which properties are accurate representations of the given function.#### Properties to Consider:1. **The graph has a zero at \( x = -2 \).**2. **The domain of the function is all real numbers.** - Rational functions can have restrictions in their domain where the denominator is zero.3. **The graph has negative values on the interval (-6, -2).**4. **The graph has a hole at (2, 0.67).** - This suggests a point of discontinuity.5. **The graph has no maximum or minimum values.**6. **The end behavior of the function is as \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \), and as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \).** - This discusses the behavior of the function as \( x \) values grow extremely large or small.7. **The graph is never increasing.**8. **The graph has no symmetry.**9. **The graph has a horizontal asymptote at \( y = \frac{1}{3} \).** - Horizontal asymptotes reflect the behavior of \( f(x) \) as \( x \rightarrow \pm \infty \).10. **The graph has a vertical asymptote at \( x = 2 \).** - Vertical asymptotes occur where the denominator is zero but not the numerator.To analyze and verify which of these properties hold true, you will need to perform algebraic manipulation, identify zeroes and asymptotes, and understand the function's behavior at extreme values.#### Detailed Explanations of the Statements:1. **Zeros**: Find where the numerator equals zero, i.e., \( x^2 + 4x - 12 = 0 \) gives the roots \( x = -6 \) and \( x = 2 \).2. **Domain**: Examine where the denominator

The equation of a rational function then we have to select…

Given the equation for the following rational function, select ALL properties that are TRUE for the graph of f(x). x²+4x-12 f(x) = 22 +42-12 3x²-12 The graph has a zero at x = -2. The domain of the function is all real numbers. | The graph has negative values on the interval (-6, -2). The graph has a hole at (2, 0.67). The graph has no maximum or minimum values. The end behavior of the function is as x → -∞, f(x) The graph is never increasing. The graph has no symmetry. The graph has a horizontal asymptote at y = The graph has a vertical asymptote at x = 2. 1/1. and as x→ ∞, f(x) ➜80.