Not substituting the correct values into the function

Oct 13, 2025 by leolavanderia.com.br 78 views

Suppose that the functions $g$ and $h$ are defined as follows:

Function Definitions

The functions $g$ and $h$ are defined as follows:

${ g(x) = x + 2 }$

${ h(x) = (x + 2)(x + 4) }$

Part (a) - Finding $\left(\frac{g}{h}\right)(-3)$

To find $\left(\frac{g}{h}\right)(-3)$ , we need to substitute $x = -3$ into the function $\frac{g(x)}{h(x)}$ .

First, let's find $g(-3)$ and $h(-3)$ :

${ g(-3) = (-3) + 2 = -1 }$

${ h(-3) = (-3 + 2)(-3 + 4) = (-1)(1) = -1 }$

Now, we can substitute these values into the function $\frac{g(x)}{h(x)}$ :

${ \left(\frac{g}{h}\right)(-3) = \frac{g(-3)}{h(-3)} = \frac{-1}{-1} = 1 }$

Therefore, $\left(\frac{g}{h}\right)(-3) = 1$ .

Part (b) - Finding all values that are NOT in the domain of $\frac{g}{h}$

To find all values that are NOT in the domain of $\frac{g}{h}$ , we need to find the values of $x$ that make $h(x) = 0$ .

Let's factor $h(x)$ :

${ h(x) = (x + 2)(x + 4) }$

We can see that $h(x) = 0$ when either $x + 2 = 0$ or $x + 4 = 0$ .

Solving for $x$ , we get:

${ x + 2 = 0 \implies x = -2 }$

${ x + 4 = 0 \implies x = -4 }$

Therefore, the values that are NOT in the domain of $\frac{g}{h}$ are $x = -2$ and $x = -4$ .

Conclusion

In this discussion, we found that $\left(\frac{g}{h}\right)(-3) = 1$ and that the values that are NOT in the domain of $\frac{g}{h}$ are $x = -2$ and $x = -4$ .

Final Answer

The final answer is $\boxed{1}$ for part (a) and $\boxed{-2, -4}$ for part (b).
Suppose that the functions $g$ and $h$ are defined as follows:

Q&A

Q: What is the definition of the function $g(x)$ ?

A: The function $g(x)$ is defined as $g(x) = x + 2$ .

Q: What is the definition of the function $h(x)$ ?

A: The function $h(x)$ is defined as $h(x) = (x + 2)(x + 4)$ .

Q: How do we find $\left(\frac{g}{h}\right)(-3)$ ?

A: To find $\left(\frac{g}{h}\right)(-3)$ , we need to substitute $x = -3$ into the function $\frac{g(x)}{h(x)}$ . We first find $g(-3)$ and $h(-3)$ , and then substitute these values into the function.

Q: What are the values of $g(-3)$ and $h(-3)$ ?

A: The values of $g(-3)$ and $h(-3)$ are $-1$ and $-1$ , respectively.

Q: What is the value of $\left(\frac{g}{h}\right)(-3)$ ?

A: The value of $\left(\frac{g}{h}\right)(-3)$ is $1$ .

Q: How do we find all values that are NOT in the domain of $\frac{g}{h}$ ?

A: To find all values that are NOT in the domain of $\frac{g}{h}$ , we need to find the values of $x$ that make $h(x) = 0$ . We can factor $h(x)$ and solve for $x$ .

Q: What are the values of $x$ that make $h(x) = 0$ ?

A: The values of $x$ that make $h(x) = 0$ are $x = -2$ and $x = -4$ .

Q: What are the values that are NOT in the domain of $\frac{g}{h}$ ?

A: The values that are NOT in the domain of $\frac{g}{h}$ are $x = -2$ and $x = -4$ .

Common Mistakes

Not substituting the correct values into the function: Make sure to substitute the correct values of $g(x)$ and $h(x)$ into the function $\frac{g(x)}{h(x)}$ .
Not factoring the function $h(x)$ : Make sure to factor the function $h(x)$ to find the values of $x$ that make $h(x) = 0$ .
Not solving for $x$ correctly: Make sure to solve for $x$ correctly when finding the values of $x$ that make $h(x) = 0$ .

Tips and Tricks

Use the correct function definitions: Make sure to use the correct function definitions for $g(x)$ and $h(x)$ .
Substitute the correct values into the function: Make sure to substitute the correct values of $g(x)$ and $h(x)$ into the function $\frac{g(x)}{h(x)}$ .
Factor the function $h(x)$ correctly: Make sure to factor the function $h(x)$ correctly to find the values of $x$ that make $h(x) = 0$ .

Conclusion

In this Q&A article, we discussed how to find $\left(\frac{g}{h}\right)(-3)$ and all values that are NOT in the domain of $\frac{g}{h}$ . We also discussed common mistakes and tips and tricks for solving these problems.