Find (f o g)(x) from f(x)= 3x and g(x)=4x²-5x+1

To find the composition \((f \circ g)(x)\), we need to evaluate \(f(g(x))\). This means we'll substitute the expression for \(g(x)\) into the function \(f(x)\).

Given:
\[
f(x) = 3x \quad \text{and} \quad g(x) = 4x^2 - 5x + 1
\]

### Step 1: Substitute \(g(x)\) into \(f(x)\)

Since \(f(x) = 3x\), we substitute \(g(x)\) in place of \(x\) in \(f(x)\):

\[
f(g(x)) = 3 \cdot g(x)
\]

Substitute \(g(x) = 4x^2 - 5x + 1\):

\[
f(g(x)) = 3 \cdot (4x^2 - 5x + 1)
\]

### Step 2: Distribute and simplify

Now, multiply the terms:

\[
f(g(x)) = 3 \cdot 4x^2 - 3 \cdot 5x + 3 \cdot 1
\]

\[
f(g(x)) = 12x^2 - 15x + 3
\]

### Conclusion

Thus, the composition \((f \circ g)(x)\) is:

\[
(f \circ g)(x) = 12x^2 - 15x + 3
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you find \((g \circ f)(x)\) with the same functions?
2. What is the derivative of \((f \circ g)(x)\) with respect to \(x\)?
3. How do you evaluate \((f \circ g)(2)\)?
4. What is the domain of \((f \circ g)(x)\)?
5. Can you explain how to determine whether a function composition is a one-to-one function?

#### Tip:
When composing functions, always carefully substitute the inner function into the outer function's formula to avoid errors.

Learn how to find the composition of two functions, (f ∘ g)(x), where f(x) = 3x and g(x) = 4x² - 5x + 1. This solution demonstrates function substitution and simplifies the expression to 12x² - 15x + 3, covering key algebraic concepts for Grades 9-10.

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