If f(x)=3x2+4x−7
 and g(x)=e3x
, find expressions for f∘g(x)
 and g∘f(x)
.
Recall: f∘g(x)≡f(g(x))
 and g∘f(x)≡g(f(x))


To find the expressions for \( f \circ g(x) \) and \( g \circ f(x) \), we need to compose the functions \( f(x) = 3x^2 + 4x - 7 \) and \( g(x) = e^{3x} \) as follows:

### 1. \( f \circ g(x) \) (which means \( f(g(x)) \)):
This composition means we will substitute \( g(x) = e^{3x} \) into \( f(x) \).

So, \( f \circ g(x) = f(g(x)) = f(e^{3x}) \).

Substitute \( x = e^{3x} \) into \( f(x) \):

\[
f(e^{3x}) = 3(e^{3x})^2 + 4(e^{3x}) - 7
\]

Simplify the expression:

\[
f(e^{3x}) = 3e^{6x} + 4e^{3x} - 7
\]

So, \( f \circ g(x) = 3e^{6x} + 4e^{3x} - 7 \).

### 2. \( g \circ f(x) \) (which means \( g(f(x)) \)):
This composition means we will substitute \( f(x) = 3x^2 + 4x - 7 \) into \( g(x) \).

So, \( g \circ f(x) = g(f(x)) = g(3x^2 + 4x - 7) \).

Substitute \( x = 3x^2 + 4x - 7 \) into \( g(x) \):

\[
g(3x^2 + 4x - 7) = e^{3(3x^2 + 4x - 7)}
\]

Simplify the expression:

\[
g(3x^2 + 4x - 7) = e^{9x^2 + 12x - 21}
\]

So, \( g \circ f(x) = e^{9x^2 + 12x - 21} \).

### Final Expressions:
- \( f \circ g(x) = 3e^{6x} + 4e^{3x} - 7 \)
- \( g \circ f(x) = e^{9x^2 + 12x - 21} \)

### Do you want any further details or have any questions?

Here are 5 related questions you might consider:
1. How would the composition \( g \circ g(x) \) look like for \( g(x) = e^{3x} \)?
2. What is the derivative of \( f \circ g(x) \) with respect to \( x \)?
3. Can you evaluate \( f \circ g(0) \) and \( g \circ f(0) \)?
4. How does the composition of functions affect their domains?
5. If \( h(x) = \sin(x) \), how would you find \( f \circ h(x) \)?

**Tip:** When composing functions, always ensure that the output of the inner function is a valid input for the outer function. This helps to avoid domain issues.

Understand how to compose functions f(x) = 3x^2 + 4x - 7 and g(x) = e^(3x). Learn step-by-step how to calculate f(g(x)) and g(f(x)), including detailed examples and simplifications. Suitable for advanced high school students and anyone studying function composition in mathematics.

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