1. Given \( f(x)=2 x+3 \) and \( g(x)=x^{2}-5 \), find the following: a. \( (f+g)(3) \) b. \( (f-g)(4) \) c. \( (f g)(-2) \) d. \( \left(\frac{f}{g}\right)(2) \)
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To solve the problem, let’s compute each part step by step. Given: \( f(x) = 2x + 3 \) \( g(x) = x^2 - 5 \) ### a. \( (f+g)(3) \) First, find \( f(3) \) and \( g(3) \): - \( f(3) = 2(3) + 3 = 6 + 3 = 9 \) - \( g(3) = 3^2 - 5 = 9 - 5 = 4 \) Now, add these: \[ (f+g)(3) = f(3) + g(3) = 9 + 4 = 13 \] ### b. \( (f-g)(4) \) Next, find \( f(4) \) and \( g(4) \): - \( f(4) = 2(4) + 3 = 8 + 3 = 11 \) - \( g(4) = 4^2 - 5 = 16 - 5 = 11 \) Now, subtract these: \[ (f-g)(4) = f(4) - g(4) = 11 - 11 = 0 \] ### c. \( (fg)(-2) \) Find \( f(-2) \) and \( g(-2) \): - \( f(-2) = 2(-2) + 3 = -4 + 3 = -1 \) - \( g(-2) = (-2)^2 - 5 = 4 - 5 = -1 \) Multiply these: \[ (fg)(-2) = f(-2) \cdot g(-2) = (-1) \cdot (-1) = 1 \] ### d. \( \left(\frac{f}{g}\right)(2) \) Find \( f(2) \) and \( g(2) \): - \( f(2) = 2(2) + 3 = 4 + 3 = 7 \) - \( g(2) = 2^2 - 5 = 4 - 5 = -1 \) Now, find the quotient: \[ \left(\frac{f}{g}\right)(2) = \frac{f(2)}{g(2)} = \frac{7}{-1} = -7 \] ### Summary of Results: a. \( (f+g)(3) = 13 \) b. \( (f-g)(4) = 0 \) c. \( (fg)(-2) = 1 \) d. \( \left(\frac{f}{g}\right)(2) = -7 \)