Math  /  Algebra

Questionf(x)=x3 and g(x)=x3+8f(x)=\sqrt[3]{x} \text { and } g(x)=x^{3}+8
Step 1 of 2: Find the formula for (fg)(x)\left(\frac{f}{g}\right)(x) and simplify your answer.

Studdy Solution

STEP 1

What is this asking? Given two functions, f(x)f(x) and g(x)g(x), find the formula for (fg)(x)\left(\frac{f}{g}\right)(x), which means dividing f(x)f(x) by g(x)g(x). Watch out! Remember that division by zero is undefined, so we need to be careful about the values of xx that make the denominator equal to zero.

STEP 2

1. Define the functions
2. Find the formula for (fg)(x)\left(\frac{f}{g}\right)(x)

STEP 3

We are given two exciting functions!
We have f(x)=x3f(x) = \sqrt[3]{x}, which is the cube root function, and g(x)=x3+8g(x) = x^3 + 8.
Let's keep these in mind as we move forward!

STEP 4

So, (fg)(x)\left(\frac{f}{g}\right)(x) just means f(x)g(x)\frac{f(x)}{g(x)}.
It's like a function party, and we're combining them through division!

STEP 5

Let's **substitute** the given expressions for f(x)f(x) and g(x)g(x) into our formula.
We get: (fg)(x)=x3x3+8 \left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x^3 + 8}

STEP 6

Now, we need to consider the values of xx that would make the denominator zero.
We set g(x)g(x) equal to zero and solve for xx: x3+8=0 x^3 + 8 = 0 x3=8 x^3 = -8 x=83 x = \sqrt[3]{-8} x=2 x = -2

STEP 7

So, xx cannot be equal to **-2**, because that would make our denominator zero, and we can't divide by zero!
It's like trying to share a pizza with zero people – it just doesn't make sense!

STEP 8

The formula for (fg)(x)\left(\frac{f}{g}\right)(x) is: (fg)(x)=x3x3+8 \left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x^3 + 8} where x2x \ne -2.

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