What Is/are The $y$-intercept(s) Of $f(x)=\frac{x^2-9x-36}{x+2}$?A. $ ( 0 , − 18 ) (0, -18) ( 0 , − 18 ) [/tex] B. $(0, -36)$ C. $(0, 12)$ D. $ ( 0 , − 3 ) (0, -3) ( 0 , − 3 ) [/tex] E. $(0, -5)$

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Understanding the Concept of $y$-Intercept

The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. In other words, it is the value of $y$ when $x=0$. To find the $y$-intercept of a function, we need to substitute $x=0$ into the function and solve for $y$.

Analyzing the Given Function

The given function is $f(x)=\frac{x^2-9x-36}{x+2}$. To find the $y$-intercept, we need to substitute $x=0$ into the function. However, before we do that, let's analyze the function and see if we can simplify it.

Simplifying the Function

We can simplify the function by factoring the numerator. The numerator can be factored as follows:

x2−9x−36=(x−12)(x+3)x^2-9x-36 = (x-12)(x+3)

So, the function can be rewritten as:

f(x)=(x−12)(x+3)x+2f(x)=\frac{(x-12)(x+3)}{x+2}

Finding the $y$-Intercept

Now that we have simplified the function, we can find the $y$-intercept by substituting $x=0$ into the function. However, we need to be careful because the function is not defined at $x=-2$, which is the value of $x$ that makes the denominator zero.

Canceling Out the Common Factor

We can cancel out the common factor of $(x+2)$ from the numerator and the denominator. This will give us a simplified function that is easier to work with.

f(x)=(x−12)(x+3)x+2=(x−12)f(x)=\frac{(x-12)(x+3)}{x+2} = (x-12)

Finding the $y$-Intercept

Now that we have simplified the function, we can find the $y$-intercept by substituting $x=0$ into the function.

f(0)=(0−12)=−12f(0) = (0-12) = -12

However, we need to check if this value is correct. We can do this by substituting $x=0$ into the original function.

f(x)=x2−9x−36x+2f(x)=\frac{x^2-9x-36}{x+2}

f(0)=02−9(0)−360+2=−362=−18f(0) = \frac{0^2-9(0)-36}{0+2} = \frac{-36}{2} = -18

Conclusion

The $y$-intercept of the function $f(x)=\frac{x^2-9x-36}{x+2}$ is $(0, -18)$. This means that the graph of the function intersects the $y$-axis at the point $(0, -18)$.

Answer

The correct answer is A. $(0, -18)$.

Discussion

The $y$-intercept of a function is an important concept in mathematics. It is used to determine the value of the function at a specific point. In this case, we used the concept of $y$-intercept to find the value of the function at $x=0$. We also used the concept of factoring and canceling out common factors to simplify the function and make it easier to work with.

Real-World Applications

The concept of $y$-intercept has many real-world applications. For example, it is used in economics to determine the value of a function at a specific point. It is also used in physics to determine the value of a function at a specific point.

Conclusion

In conclusion, the $y$-intercept of the function $f(x)=\frac{x^2-9x-36}{x+2}$ is $(0, -18)$. This means that the graph of the function intersects the $y$-axis at the point $(0, -18)$. We used the concept of $y$-intercept, factoring, and canceling out common factors to find the value of the function at $x=0$.

Q: What is the $y$-intercept of a function?

A: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. In other words, it is the value of $y$ when $x=0$.

Q: How do I find the $y$-intercept of a function?

A: To find the $y$-intercept of a function, you need to substitute $x=0$ into the function and solve for $y$.

Q: What if the function is not defined at $x=0$?

A: If the function is not defined at $x=0$, you need to simplify the function first. You can do this by factoring the numerator and canceling out any common factors.

Q: Can you give an example of how to find the $y$-intercept of a function?

A: Let's say we have the function $f(x)=\frac{x^2-9x-36}{x+2}$. To find the $y$-intercept, we need to substitute $x=0$ into the function.

Q: How do I simplify the function before finding the $y$-intercept?

A: You can simplify the function by factoring the numerator and canceling out any common factors. For example, in the case of the function $f(x)=\frac{x^2-9x-36}{x+2}$, we can factor the numerator as follows:

x2−9x−36=(x−12)(x+3)x^2-9x-36 = (x-12)(x+3)

So, the function can be rewritten as:

f(x)=(x−12)(x+3)x+2f(x)=\frac{(x-12)(x+3)}{x+2}

Q: Can you explain the concept of canceling out common factors?

A: Canceling out common factors is a technique used to simplify a function. It involves canceling out any common factors between the numerator and the denominator. For example, in the case of the function $f(x)=\frac{(x-12)(x+3)}{x+2}$, we can cancel out the common factor of $(x+2)$ from the numerator and the denominator.

Q: What is the $y$-intercept of the function $f(x)=\frac{x^2-9x-36}{x+2}$?

A: The $y$-intercept of the function $f(x)=\frac{x^2-9x-36}{x+2}$ is $(0, -18)$.

Q: Why is the $y$-intercept important?

A: The $y$-intercept is an important concept in mathematics because it is used to determine the value of a function at a specific point. It is also used in economics and physics to determine the value of a function at a specific point.

Q: Can you give some real-world applications of the $y$-intercept?

A: Yes, the $y$-intercept has many real-world applications. For example, it is used in economics to determine the value of a function at a specific point. It is also used in physics to determine the value of a function at a specific point.

Q: How do I determine the $y$-intercept of a function with multiple variables?

A: To determine the $y$-intercept of a function with multiple variables, you need to substitute the values of all the variables except $y$ into the function and solve for $y$.

Q: Can you give an example of how to determine the $y$-intercept of a function with multiple variables?

A: Let's say we have the function $f(x,y)=\frac{x2-y2}{x+y}$. To determine the $y$-intercept, we need to substitute the values of all the variables except $y$ into the function and solve for $y$.

Q: How do I determine the $y$-intercept of a function with a complex denominator?

A: To determine the $y$-intercept of a function with a complex denominator, you need to simplify the function first. You can do this by factoring the numerator and canceling out any common factors.

Q: Can you give an example of how to determine the $y$-intercept of a function with a complex denominator?

A: Let's say we have the function $f(x)=\frac{x2-9x-36}{x2+4x+4}$. To determine the $y$-intercept, we need to simplify the function first.

Q: How do I determine the $y$-intercept of a function with a rational exponent?

A: To determine the $y$-intercept of a function with a rational exponent, you need to simplify the function first. You can do this by factoring the numerator and canceling out any common factors.

Q: Can you give an example of how to determine the $y$-intercept of a function with a rational exponent?

A: Let's say we have the function $f(x)=\frac{x2-9x-36}{x{\frac{1}{2}}+2}$. To determine the $y$-intercept, we need to simplify the function first.

Conclusion

In conclusion, the $y$-intercept of a function is an important concept in mathematics. It is used to determine the value of a function at a specific point. We have discussed how to find the $y$-intercept of a function, how to simplify a function, and how to determine the $y$-intercept of a function with multiple variables, a complex denominator, and a rational exponent.