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URL: https://nakafa.com/en/subjects/mathematics/function-composition-inverse-function/multiplication-division-function
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Learn function multiplication and division, including domain rules, zero restrictions, and examples for (f·g)(x) and (f/g)(x).
---
## Multiplication of Two Functions
Multiplying two functions, $$f$$ and $$g$$, is as easy as multiplying two numbers. We just multiply the result of $$f(x)$$ by $$g(x)$$ for the same value of $$x$$. The result is a new function $$(f \cdot g)$$.
Visible text: Multiplying two functions, and , is as easy as multiplying two numbers. We just multiply the result of by for the same value of . The result is a new function .
Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Function Multiplication Visualization
- description: Observe how the lines $$f(x)=x$$ and{" "}
$$g(x)=2$$ are multiplied to become{" "}
$$(f \cdot g)(x)=2x$$.
Visible text: Observe how the lines and{" "}
are multiplied to become{" "}
.
- data: [
{
points: Array.from({ length: 11 }, (_, i) => ({
x: i - 5,
y: i - 5,
z: 0,
})),
color: getColor("ORANGE"),
labels: [{ text: "f(x)=x", at: 6, offset: [1, -0.5, 0] }],
},
{
points: Array.from({ length: 11 }, (_, i) => ({ x: i - 5, y: 2, z: 0 })),
color: getColor("SKY"),
labels: [{ text: "g(x)=2", at: 8, offset: [1, 0.5, 0] }],
},
{
points: Array.from({ length: 11 }, (_, i) => ({
x: i - 5,
y: 2 * (i - 5),
z: 0,
})),
color: getColor("ROSE"),
labels: [{ text: "(f⋅g)(x)=2x", at: 7, offset: [-1.5, 1, 0] }],
},
]
```math
(f \cdot g)(x) = f(x) \cdot g(x)
```
Just like addition and subtraction, this multiplication machine $$(f \cdot g)$$ can only process raw materials (values of $$x$$) that can be processed by _both_ original machines, $$f$$ and $$g$$. So, its domain is the intersection of the domain of $$f$$ and the domain of $$g$$.
Visible text: Just like addition and subtraction, this multiplication machine can only process raw materials (values of ) that can be processed by _both_ original machines, and . So, its domain is the intersection of the domain of and the domain of .
```math
D_{f \cdot g} = D_f \cap D_g
```
### Example of Multiplication
Let's use slightly different functions this time:
1. $$f(x) = x^2$$, with domain $$D_f = \{x | x \in \mathbb{R}\}$$ (all
real numbers).
2. $$g(x) = x - 1$$, with domain $$D_g = \{x | x \in \mathbb{R}\}$$ (all
real numbers).
Visible text: 1. , with domain (all
real numbers).
2. , with domain (all
real numbers).
**Step** $$1$$: Determine the resulting function from multiplication
Visible text: **Step** : Determine the resulting function from multiplication
Component: MathContainer
Children:
```math
(f \cdot g)(x) = f(x) \cdot g(x)
```
```math
= (x^2) \cdot (x - 1)
```
```math
= x^3 - x^2
```
**Step** $$2$$: Determine the domain of the resulting function
Visible text: **Step** : Determine the domain of the resulting function
We find the intersection of $$D_f$$ and $$D_g$$:
Visible text: We find the intersection of and :
Component: MathContainer
Children:
```math
D_{f \cdot g} = D_f \cap D_g
```
```math
= \{x | x \in \mathbb{R}\} \cap \{x | x \in \mathbb{R}\}
```
```math
= \{x | x \in \mathbb{R}\}
```
So, the resulting function from multiplication is $$(f \cdot g)(x) = x^3 - x^2$$ with the domain of all real numbers.
Visible text: So, the resulting function from multiplication is with the domain of all real numbers.
## Division of Two Functions
Dividing function $$f$$ by function $$g$$ is also similar: we divide the result of $$f(x)$$ by $$g(x)$$. The result is a new function $$(\frac{f}{g})$$.
Visible text: Dividing function by function is also similar: we divide the result of by . The result is a new function .
```math
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
```
Now, there's a **very important additional rule!** We know that division by zero is not allowed. So, besides the value of $$x$$ needing to be in the domain of both $$f$$ and $$g$$, the value of $$g(x)$$ (the divisor function) **cannot be equal to zero**.
Visible text: Now, there's a **very important additional rule!** We know that division by zero is not allowed. So, besides the value of needing to be in the domain of both and , the value of (the divisor function) **cannot be equal to zero**.
Therefore, the domain of the division function $$(\frac{f}{g})$$ is the intersection of domains $$D_f$$ and $$D_g$$, but we must _exclude_ all values of $$x$$ that cause $$g(x) = 0$$.
Visible text: Therefore, the domain of the division function is the intersection of domains and , but we must _exclude_ all values of that cause .
```math
D_{\frac{f}{g}} = D_f \cap D_g - \{x | g(x) = 0\}
```
The $$-$$ sign here means "minus" or "excluded".
Visible text: The sign here means "minus" or "excluded".
### Example of Division
We use the same functions as in the multiplication example:
1. $$f(x) = x^2$$, $$D_f = \{x | x \in \mathbb{R}\}$$
2. $$g(x) = x - 1$$, $$D_g = \{x | x \in \mathbb{R}\}$$
Visible text: 1. ,
2. ,
**Step** $$1$$: Determine the resulting function from division
Visible text: **Step** : Determine the resulting function from division
```math
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2}{x-1}
```
**Step** $$2$$: Determine the domain of the resulting function
Visible text: **Step** : Determine the domain of the resulting function
First, find the intersection of $$D_f$$ and $$D_g$$:
Visible text: First, find the intersection of and :
```math
D_f \cap D_g = \{x | x \in \mathbb{R}\}
```
Second, find the value of $$x$$ that makes $$g(x) = 0$$:
Visible text: Second, find the value of that makes :
Component: MathContainer
Children:
```math
g(x) = 0
```
```math
x - 1 = 0
```
```math
x = 1
```
Third, exclude the value $$x=1$$ from the intersection of the domains:
Visible text: Third, exclude the value from the intersection of the domains:
```math
D_{\frac{f}{g}} = \{x | x \in \mathbb{R}\} - \{1\}
```
Or it can also be written as:
```math
D_{\frac{f}{g}} = \{x | x \in \mathbb{R}, x \neq 1\}
```
So, the resulting function from division is $$(\frac{f}{g})(x) = \frac{x^2}{x-1}$$ with the domain of all real numbers except $$x=1$$.
Visible text: So, the resulting function from division is with the domain of all real numbers except .
## Practice Problems
Given the function $$f(x) = \sqrt{x+4}$$ with $$D_f = \{x | x \ge -4, x \in \mathbb{R}\}$$ and function $$g(x) = x^2 - 9$$ with $$D_g = \{x | x \in \mathbb{R}\}$$.
Visible text: Given the function with and function with .
1. Determine $$(f \cdot g)(x)$$ and its domain $$D_{f \cdot g}$$.
2. Determine $$(\frac{f}{g})(x)$$ and its domain $$D_{\frac{f}{g}}$$.
3. Calculate the value of $$(f \cdot g)(5)$$.
4. Is $$(\frac{f}{g})(3)$$ defined? Explain.
Visible text: 1. Determine and its domain .
2. Determine and its domain .
3. Calculate the value of .
4. Is defined? Explain.
### Answer Key
1. **Finding $$(f \cdot g)(x)$$:**
```math
(f \cdot g)(x) = f(x) \cdot g(x)
```
```math
= (\sqrt{x+4}) \cdot (x^2 - 9)
```
```math
= (x^2 - 9)\sqrt{x+4}
```
**Finding Domain $$D_{f \cdot g}$$:**
```math
D_{f \cdot g} = D_f \cap D_g
```
```math
= \{x | x \ge -4, x \in \mathbb{R}\} \cap \{x | x \in \mathbb{R}\}
```
```math
= \{x | x \ge -4, x \in \mathbb{R}\}
```
So, $$(f \cdot g)(x) = (x^2 - 9)\sqrt{x+4}$$ with domain $$\{x | x \ge -4, x \in \mathbb{R}\}$$.
2. **Finding $$(\frac{f}{g})(x)$$:**
```math
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x+4}}{x^2 - 9}
```
**Finding Domain $$D_{\frac{f}{g}}$$:**
Domain intersection: $$D_f \cap D_g = \{x | x \ge -4, x \in \mathbb{R}\}$$.
Find $$x$$ that makes $$g(x)=0$$:
```math
g(x) = 0
```
```math
x^2 - 9 = 0
```
```math
(x-3)(x+3) = 0
```
```math
x = 3 \text{ or} x = -3
```
Exclude $$x=3$$ and $$x=-3$$ from the domain intersection:
```math
D_{\frac{f}{g}} = \{x | x \ge -4, x \in \mathbb{R}\} - \{-3, 3\}
```
Or it can be written as:
```math
D_{\frac{f}{g}} = \{x | x \ge -4, x \in \mathbb{R}, x \neq -3, x \neq 3\}
```
So, $$(\frac{f}{g})(x) = \frac{\sqrt{x+4}}{x^2 - 9}$$ with domain $$\{x | x \ge -4, x \neq -3, x \neq 3\}$$.
3. **Calculating $$(f \cdot g)(5)$$:**
We use the result from number $$1$$: $$(f \cdot g)(x) = (x^2 - 9)\sqrt{x+4}$$.
Since $$5 \ge -4$$, $$x=5$$ is in the domain $$D_{f \cdot g}$$.
```math
(f \cdot g)(5) = (5^2 - 9)\sqrt{5+4}
```
```math
= (25 - 9)\sqrt{9}
```
```math
= (16)(3)
```
```math
= 48
```
4. **Is $$(\frac{f}{g})(3)$$ defined?**
Undefined. We look at the domain of $$(\frac{f}{g})(x)$$ from number $$2$$, which is $$\{x | x \ge -4, x \neq -3, x \neq 3\}$$. The value $$x=3$$ is explicitly excluded from the domain because it would cause the denominator $$g(x) = x^2 - 9$$ to become zero ($$3^2 - 9 = 0$$). Division by zero is not allowed in mathematics.
Visible text: 1. **Finding :**
**Finding Domain :**
So, with domain .
2. **Finding :**
**Finding Domain :**
Domain intersection: .
Find that makes :
Exclude and from the domain intersection:
Or it can be written as:
So, with domain .
3. **Calculating :**
We use the result from number : .
Since , is in the domain .
4. **Is defined?**
Undefined. We look at the domain of from number , which is . The value is explicitly excluded from the domain because it would cause the denominator to become zero (). Division by zero is not allowed in mathematics.