Math / Algebra

Question $f(x)=x^{\frac{3}{2}} \text { and } g(x)=-2 x$ of 2: Find the formula for $\left(\frac{f}{g}\right)(x)$ and simplify your answer.

Studdy Solution

STEP 1

What is this asking? We're taking two functions, $f(x)$ and $g(x)$ , and we need to find a brand new function that's $f(x)$ divided by $g(x)$ ! Watch out! Don't forget to consider what happens when $g(x)$ is zero – we can't divide by zero!

STEP 2

1. Define the functions
2. Build the combined function
3. Simplify the combined function

STEP 3

Alright, let's define our functions!
We have $f(x) = x^{\frac{3}{2}}$ and $g(x) = -2x$ .
Super easy so far!

STEP 4

Now, let's build our new function $\left(\frac{f}{g}\right)(x)$ .
This just means we take $f(x)$ and divide it by $g(x)$ .
So, we have: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^{\frac{3}{2}}}{-2x}$

STEP 5

Time to simplify!
Remember, $\frac{x^a}{x^b} = x^{a-b}$ .
In our case, we have $x$ raised to the power of $\frac{3}{2}$ in the numerator and $x$ raised to the power of $1$ in the denominator.

STEP 6

So, we subtract the exponents: $\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}$ .

STEP 7

This gives us: $\frac{x^{\frac{3}{2}}}{-2x} = \frac{x^{\frac{3}{2}}}{-2 \cdot x^1} = \frac{1}{-2} \cdot \frac{x^{\frac{3}{2}}}{x^1} = \frac{1}{-2} \cdot x^{\frac{3}{2} - 1} = -\frac{1}{2} x^{\frac{1}{2}}$

STEP 8

Remember we can't divide by zero, so $g(x) = -2x$ cannot be zero.
This means $x$ itself cannot be zero!
So, we must specify that $x \ne 0$ .

STEP 9

Our simplified function is $\left(\frac{f}{g}\right)(x) = -\frac{1}{2} x^{\frac{1}{2}}$ where $x \ne 0$ .

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Question $f(x)=x^{\frac{3}{2}} \text { and } g(x)=-2 x$ of 2: Find the formula for $\left(\frac{f}{g}\right)(x)$ and simplify your answer.

Studdy Solution

STEP 1

What is this asking? We're taking two functions, $f(x)$ and $g(x)$ , and we need to find a brand new function that's $f(x)$ divided by $g(x)$ ! Watch out! Don't forget to consider what happens when $g(x)$ is zero – we can't divide by zero!

STEP 2

1. Define the functions
2. Build the combined function
3. Simplify the combined function

STEP 3

Alright, let's define our functions!
We have $f(x) = x^{\frac{3}{2}}$ and $g(x) = -2x$ .
Super easy so far!

STEP 4

Now, let's build our new function $\left(\frac{f}{g}\right)(x)$ .
This just means we take $f(x)$ and divide it by $g(x)$ .
So, we have: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^{\frac{3}{2}}}{-2x}$

STEP 5

Time to simplify!
Remember, $\frac{x^a}{x^b} = x^{a-b}$ .
In our case, we have $x$ raised to the power of $\frac{3}{2}$ in the numerator and $x$ raised to the power of $1$ in the denominator.

STEP 6

So, we subtract the exponents: $\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}$ .

STEP 7

This gives us: $\frac{x^{\frac{3}{2}}}{-2x} = \frac{x^{\frac{3}{2}}}{-2 \cdot x^1} = \frac{1}{-2} \cdot \frac{x^{\frac{3}{2}}}{x^1} = \frac{1}{-2} \cdot x^{\frac{3}{2} - 1} = -\frac{1}{2} x^{\frac{1}{2}}$

STEP 8

Remember we can't divide by zero, so $g(x) = -2x$ cannot be zero.
This means $x$ itself cannot be zero!
So, we must specify that $x \ne 0$ .

STEP 9

Our simplified function is $\left(\frac{f}{g}\right)(x) = -\frac{1}{2} x^{\frac{1}{2}}$ where $x \ne 0$ .

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Questionf(x)=x32 and g(x)=−2xf(x)=x^{\frac{3}{2}} \text { and } g(x)=-2 xf(x)=x23​ and g(x)=−2x of 2: Find the formula for (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x) and simplify your answer.

Studdy Solution

STEP 1

What is this asking? We're taking two functions, f(x)f(x)f(x) and g(x)g(x)g(x), and we need to find a *brand new* function that's f(x)f(x)f(x) divided by g(x)g(x)g(x)! Watch out! Don't forget to consider what happens when g(x)g(x)g(x) is zero – we can't divide by zero!

STEP 2

1. Define the functions2. Build the combined function3. Simplify the combined function

STEP 3

Alright, let's **define** our functions!We have f(x)=x32f(x) = x^{\frac{3}{2}}f(x)=x23​ and g(x)=−2xg(x) = -2xg(x)=−2x.Super easy so far!

STEP 4

STEP 5

Time to **simplify**!Remember, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}xbxa​=xa−b.In our case, we have xxx raised to the power of 32\frac{3}{2}23​ in the numerator and xxx raised to the power of 111 in the denominator.

STEP 6

So, we **subtract** the exponents: 32−1=32−22=12\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}23​−1=23​−22​=21​.

STEP 7

STEP 8

Remember we can't divide by zero, so g(x)=−2xg(x) = -2xg(x)=−2x cannot be zero.This means xxx itself cannot be zero!So, we must specify that x≠0x \ne 0x=0.

STEP 9

Our simplified function is (fg)(x)=−12x12\left(\frac{f}{g}\right)(x) = -\frac{1}{2} x^{\frac{1}{2}}(gf​)(x)=−21​x21​ where x≠0x \ne 0x=0.

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Question $f(x)=x^{\frac{3}{2}} \text { and } g(x)=-2 x$ of 2: Find the formula for $\left(\frac{f}{g}\right)(x)$ and simplify your answer.

What is this asking? We're taking two functions, $f(x)$ and $g(x)$ , and we need to find a brand new function that's $f(x)$ divided by $g(x)$ ! Watch out! Don't forget to consider what happens when $g(x)$ is zero – we can't divide by zero!

1. Define the functions
2. Build the combined function
3. Simplify the combined function

Alright, let's define our functions!
We have $f(x) = x^{\frac{3}{2}}$ and $g(x) = -2x$ .
Super easy so far!

Time to simplify!
Remember, $\frac{x^a}{x^b} = x^{a-b}$ .
In our case, we have $x$ raised to the power of $\frac{3}{2}$ in the numerator and $x$ raised to the power of $1$ in the denominator.

So, we subtract the exponents: $\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}$ .

Remember we can't divide by zero, so $g(x) = -2x$ cannot be zero.
This means $x$ itself cannot be zero!
So, we must specify that $x \ne 0$ .

Our simplified function is $\left(\frac{f}{g}\right)(x) = -\frac{1}{2} x^{\frac{1}{2}}$ where $x \ne 0$ .