Math  /  Algebra

QuestionThis problem tests calculating new functions from old ones. From the table below calculate the quantities asked for: \begin{tabular}{|ccccccc|} \hlinexx & 22 & 1 & -2 & -68 & 2 & 3 \\ f(x)f(x) & 10187 & 2 & -13 & -319123 & 7 & 22 \\ g(x)g(x) & -22241 & -2 & 7 & 619549 & -21 & -68 \\ \hline \end{tabular} g(f(3))=(fg)(1)=(f+g)(3)=\begin{array}{l} g(f(3))=\square \\ (f g)(1)=\square \\ (f+g)(3)=\square \end{array}

Studdy Solution

STEP 1

What is this asking? Given a table of values for functions f(x)f(x) and g(x)g(x), we need to find the values of some combined functions. Watch out! Don't mix up function composition with function multiplication!
Also, remember that (f+g)(x)(f+g)(x) means f(x)+g(x)f(x) + g(x).

STEP 2

1. Calculate g(f(3))g(f(3))
2. Calculate (fg)(1)(f g)(1)
3. Calculate (f+g)(3)(f+g)(3)

STEP 3

Let's **start** with the inside function, f(3)f(3).
Looking at the table, when x=3x = 3, f(x)=22f(x) = 22.
So, f(3)=22f(3) = \mathbf{22}.

STEP 4

Now we need to find g(f(3))g(f(3)), which we now know is the same as g(22)g(\mathbf{22}).
From the table, when x=22x = 22, g(x)=22241g(x) = -22241.
Therefore, g(f(3))=g(22)=22241g(f(3)) = g(22) = \mathbf{-22241}.

STEP 5

Remember that (fg)(x)(fg)(x) means f(x)g(x)f(x) \cdot g(x).
So, we need to find f(1)f(1) and g(1)g(1) and **multiply** them together.

STEP 6

From the table, f(1)=2f(1) = \mathbf{2} and g(1)=2g(1) = \mathbf{-2}.

STEP 7

Now, let's **multiply** these values: (fg)(1)=f(1)g(1)=2(2)=4(fg)(1) = f(1) \cdot g(1) = 2 \cdot (-2) = \mathbf{-4}.

STEP 8

(f+g)(3)(f+g)(3) means we need to find f(3)f(3) and g(3)g(3) and **add** them together.

STEP 9

From the table, f(3)=22f(3) = \mathbf{22} and g(3)=68g(3) = \mathbf{-68}.

STEP 10

Let's **add** these values: (f+g)(3)=f(3)+g(3)=22+(68)=46(f+g)(3) = f(3) + g(3) = 22 + (-68) = \mathbf{-46}.

STEP 11

g(f(3))=22241(fg)(1)=4(f+g)(3)=46\begin{array}{l} g(f(3)) = -22241 \\ (f g)(1) = -4 \\ (f+g)(3) = -46 \end{array}

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