How do you find the composition of two functions, f(x) = 3x^2 + 2 and g(x) = 5x^2 + 1? Show each step of your work.

Finding the Composition of Functions

To find the composition of two functions, we denote it as (f ∘ g)(x), which means we will substitute g(x) into f(x).

Step 1: Identify the Functions

We have the following two functions:

• f(x) = 3x² + 2
• g(x) = 5x² + 1

Step 2: Substitute g(x) into f(x)

We want to find (f ∘ g)(x) = f(g(x)). This means we will plug g(x) into f(x).

So:

• g(x) = 5x² + 1

Substituting g(x) into f(x):

f(g(x)) = f(5x² + 1) = 3(5x² + 1)² + 2

Step 3: Expand the Expression

Now we need to expand (5x² + 1)²:

(5x² + 1)² = 25x^4 + 10x² + 1

Next, we substitute this back into f(g(x)):

f(g(x)) = 3(25x⁴ + 10x² + 1) + 2

Step 4: Distribute the 3

Now we distribute 3 into the expanded expression:

3(25x⁴) + 3(10x²) + 3(1) + 2 = 75x⁴ + 30x² + 3 + 2

Step 5: Combine Like Terms

Finally, combine the constants:

75x⁴ + 30x² + 5

Final Result

The composition of the functions f and g is:

(f ∘ g)(x) = 75x⁴ + 30x² + 5

And that’s how you find the composition of the functions step by step!