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Question

Part 1
If ​f(x)equalsx squared minus 2 x and ​g(x)equals9 minus x cubed​, evaluate the following.
a.​ (fplus​g)(negative 2​)      b.​ (gminus​f)(negative 4​)      c.​ (ftimes​g)(negative 1​)      d. left parenthesis StartFraction g Over f EndFraction right parenthesis
​(negative 3​)
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Part 1
a.​ (fplus​g)(negative 2​)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To evaluate the functions \( f(x) \) and \( g(x) \), we first need the definitions of both functions:

• \( f(x) = x^2 - 2x \)
• \( g(x) = 9 - x^3 \)

Now, let's calculate each part:

Part a: \((f + g)(-2)\)

1. Calculate \( f(-2) \): \[ f(-2) = (-2)^2 - 2(-2) = 4 + 4 = 8 \]

2. Calculate \( g(-2) \): \[ g(-2) = 9 - (-2)^3 = 9 - (-8) = 9 + 8 = 17 \]

3. Combine \( f(-2) \) and \( g(-2) \): \[ (f + g)(-2) = f(-2) + g(-2) = 8 + 17 = 25 \]

So, the answer for part a is: \[ \boxed{25} \]

Part b: \((g - f)(-4)\)

1. Calculate \( g(-4) \): \[ g(-4) = 9 - (-4)^3 = 9 - (-64) = 9 + 64 = 73 \]

2. Calculate \( f(-4) \): \[ f(-4) = (-4)^2 - 2(-4) = 16 + 8 = 24 \]

3. Combine \( g(-4) \) and \( f(-4) \): \[ (g - f)(-4) = g(-4) - f(-4) = 73 - 24 = 49 \]

So, the answer for part b is: \[ \boxed{49} \]

Part c: \((f \cdot g)(-1)\)

1. Calculate \( f(-1) \): \[ f(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3 \]

2. Calculate \( g(-1) \): \[ g(-1) = 9 - (-1)^3 = 9 - (-1) = 9 + 1 = 10 \]

3. Combine \( f(-1) \) and \( g(-1) \): \[ (f \cdot g)(-1) = f(-1) \cdot g(-1) = 3 \cdot 10 = 30 \]

So, the answer for part c is: \[ \boxed{30} \]

Part d: \(\left(\frac{g}{f}\right)(-3)\)

1. Calculate \( g(-3) \): \[ g(-3) = 9 - (-3)^3 = 9 - (-27) = 9 + 27 = 36 \]

2. Calculate \( f(-3) \): \[ f(-3) = (-3)^2 - 2(-3) = 9 + 6 = 15 \]

3. Combine \( g(-3) \) and \( f(-3) \): \[ \left(\frac{g}{f}\right)(-3) = \frac{g(-3)}{f(-3)} = \frac{36}{15} = \frac{12}{5} \]

So, the answer for part d is: \[ \boxed{\frac{12}{5}} \]

Now we have the answers to all parts!