Clarke Bates
12/16/2023 · High School
Find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) and the domain of each. \[ f(x)=x+3, g(x)=2 x^{2}-5 x-3 \]
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Tutor-Verified Answer
Step-by-step Solution
To find \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we will first compute each composition of functions and then determine their domains.
### Step 1: Compute \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \).
1. **Find \( g(x) \)**:
\[
g(x) = 2x^2 - 5x - 3
\]
2. **Substitute \( g(x) \) into \( f(x) \)**:
\[
f(g(x)) = f(2x^2 - 5x - 3) = (2x^2 - 5x - 3) + 3
\]
Simplifying this:
\[
f(g(x)) = 2x^2 - 5x - 3 + 3 = 2x^2 - 5x
\]
Thus, we have:
\[
(f \circ g)(x) = 2x^2 - 5x
\]
### Step 2: Determine the domain of \( (f \circ g)(x) \)
The function \( g(x) = 2x^2 - 5x - 3 \) is a polynomial, which is defined for all real numbers. Therefore, the output of \( g(x) \) can be any real number, and since \( f(x) = x + 3 \) is also defined for all real numbers, the domain of \( (f \circ g)(x) \) is:
\[
\text{Domain of } (f \circ g)(x) = \mathbb{R}
\]
### Step 3: Compute \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means we will substitute \( f(x) \) into \( g(x) \).
1. **Find \( f(x) \)**:
\[
f(x) = x + 3
\]
2. **Substitute \( f(x) \) into \( g(x) \)**:
\[
g(f(x)) = g(x + 3) = 2(x + 3)^2 - 5(x + 3) - 3
\]
Expanding \( (x + 3)^2 \):
\[
(x + 3)^2 = x^2 + 6x + 9
\]
Now substituting this back into \( g \):
\[
g(f(x)) = 2(x^2 + 6x + 9) - 5(x + 3) - 3
\]
Distributing:
\[
= 2x^2 + 12x + 18 - 5x - 15 - 3
\]
Combining like terms:
\[
= 2x^2 + (12x - 5x) + (18 - 15 - 3) = 2x^2 + 7x
\]
Thus, we have:
\[
(g \circ f)(x) = 2x^2 + 7x
\]
### Step 4: Determine the domain of \( (g \circ f)(x) \)
The function \( f(x) = x + 3 \) is defined for all real numbers, and since \( g(x) = 2x^2 - 5x - 3 \) is also defined for all real numbers, the domain of \( (g \circ f)(x) \) is:
\[
\text{Domain of } (g \circ f)(x) = \mathbb{R}
\]
### Summary
The results are:
\[
(f \circ g)(x) = 2x^2 - 5x, \quad \text{Domain: } \mathbb{R}
\]
\[
(g \circ f)(x) = 2x^2 + 7x, \quad \text{Domain: } \mathbb{R}
\]
Quick Answer
\[
(f \circ g)(x) = 2x^2 - 5x, \quad \text{Domain: } \mathbb{R}
\]
\[
(g \circ f)(x) = 2x^2 + 7x, \quad \text{Domain: } \mathbb{R}
\]
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