Hilton Wells
03/26/2024 · Middle School

1. \( f(x)=2 x^{2}+3 x-4 \) and \( g(x)=2 x+3 \), Find the \( (f+g)(x) \) and \( (g+f)(x) \)

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Step-by-step Solution

To find \( (f+g)(x) \) and \( (g+f)(x) \), we need to add the functions \( f(x) \) and \( g(x) \). Given: \[ f(x) = 2x^2 + 3x - 4 \] \[ g(x) = 2x + 3 \] ### Finding \( (f+g)(x) \) \[ (f+g)(x) = f(x) + g(x) \] Substituting the expressions for \( f(x) \) and \( g(x) \): \[ (f+g)(x) = (2x^2 + 3x - 4) + (2x + 3) \] Now, combine like terms: \[ = 2x^2 + 3x + 2x - 4 + 3 \] \[ = 2x^2 + (3x + 2x) + (-4 + 3) \] \[ = 2x^2 + 5x - 1 \] ### Finding \( (g+f)(x) \) \[ (g+f)(x) = g(x) + f(x) \] Substituting the expressions for \( g(x) \) and \( f(x) \): \[ (g+f)(x) = (2x + 3) + (2x^2 + 3x - 4) \] Now, combine like terms: \[ = 2x + 3 + 2x^2 + 3x - 4 \] \[ = 2x^2 + (2x + 3x) + (3 - 4) \] \[ = 2x^2 + 5x - 1 \] ### Conclusion Both \( (f+g)(x) \) and \( (g+f)(x) \) yield the same result: \[ (f+g)(x) = 2x^2 + 5x - 1 \] \[ (g+f)(x) = 2x^2 + 5x - 1 \]

Quick Answer

\( (f+g)(x) = 2x^2 + 5x - 1 \) \( (g+f)(x) = 2x^2 + 5x - 1 \)
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