Young Mcdonald
04/06/2023 · Senior High School
2. \( f(x)=2 x^{2}+3 x-4 \) and \( g(x)=2 x+3 \), find the \( (f-g)(x) \) 3. \( f(x)=3 x-5 \) and \( g(x)=x \), Find the \( (f \bullet g)(x) \) 4. \( f(x)=3 x^{2}+4 x-3 \) and \( g(x)=x \), find the \( (f / g)(x) \)
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Step-by-step Solution
Let's solve each of the problems step by step.
### Problem 2: Find \( (f-g)(x) \)
Given:
- \( f(x) = 2x^2 + 3x - 4 \)
- \( g(x) = 2x + 3 \)
To find \( (f-g)(x) \), we subtract \( g(x) \) from \( f(x) \):
\[
(f-g)(x) = f(x) - g(x)
\]
Substituting the expressions for \( f(x) \) and \( g(x) \):
\[
(f-g)(x) = (2x^2 + 3x - 4) - (2x + 3)
\]
Now, distribute the negative sign:
\[
(f-g)(x) = 2x^2 + 3x - 4 - 2x - 3
\]
Combine like terms:
\[
(f-g)(x) = 2x^2 + (3x - 2x) + (-4 - 3)
\]
\[
(f-g)(x) = 2x^2 + x - 7
\]
### Problem 3: Find \( (f \bullet g)(x) \)
Given:
- \( f(x) = 3x - 5 \)
- \( g(x) = x \)
To find \( (f \bullet g)(x) \), we multiply \( f(x) \) and \( g(x) \):
\[
(f \bullet g)(x) = f(x) \cdot g(x)
\]
Substituting the expressions for \( f(x) \) and \( g(x) \):
\[
(f \bullet g)(x) = (3x - 5) \cdot x
\]
Distributing \( x \):
\[
(f \bullet g)(x) = 3x^2 - 5x
\]
### Problem 4: Find \( (f / g)(x) \)
Given:
- \( f(x) = 3x^2 + 4x - 3 \)
- \( g(x) = x \)
To find \( (f / g)(x) \), we divide \( f(x) \) by \( g(x) \):
\[
(f / g)(x) = \frac{f(x)}{g(x)}
\]
Substituting the expressions for \( f(x) \) and \( g(x) \):
\[
(f / g)(x) = \frac{3x^2 + 4x - 3}{x}
\]
Now, we can simplify this expression by dividing each term in the numerator by \( x \):
\[
(f / g)(x) = \frac{3x^2}{x} + \frac{4x}{x} - \frac{3}{x}
\]
\[
(f / g)(x) = 3x + 4 - \frac{3}{x}
\]
### Summary of Results
1. \( (f-g)(x) = 2x^2 + x - 7 \)
2. \( (f \bullet g)(x) = 3x^2 - 5x \)
3. \( (f / g)(x) = 3x + 4 - \frac{3}{x} \)
Quick Answer
1. \( (f-g)(x) = 2x^2 + x - 7 \)
2. \( (f \bullet g)(x) = 3x^2 - 5x \)
3. \( (f / g)(x) = 3x + 4 - \frac{3}{x} \)
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