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Given Information:

1. The linear function $f(x) = 10x - 2$ and exponential function $g(x) = 3^x - 1 + 4$.
2. The function $f(x)$ is transformed by the translation $T = \binom{5}{5}$, resulting in $f'(x)$.
3. The function $g(x)$ is transformed by the translation $T = \binom{2}{2}$, resulting in $g'(x)$.
4. Translation $T = \binom{6}{-1}$ is applied to the function $f(x)$ to form the function $h(x)$.

Now, let's go through the questions:

Question 1:

The transformation of $f(x)$ by the translation $T = \binom{5}{5}$.

The transformation involves translating the function vertically by 5 units and horizontally by 5 units. So, we apply the transformation to $f(x)$ as follows: $f'(x) = f(x - 5) + 5$ Substitute $f(x) = 10x - 2$ into the above equation: $f'(x) = 10(x - 5) - 2 + 5$ $f'(x) = 10x - 50 - 2 + 5$ $f'(x) = 10x - 47$ Thus, the correct answer for question 1 is b. $f'(x) = 10x - 47$.

Question 2:

Evaluate $f'(x)$ for specific values of $x$.

From the previous transformation, we know $f'(x) = 10x - 47$.

• $f'(2) = 10(2) - 47 = 20 - 47 = -27$
• $f'(3) = 10(3) - 47 = 30 - 47 = -17$
• $f'(4) = 10(4) - 47 = 40 - 47 = -7$
• $f'(5) = 10(5) - 47 = 50 - 47 = 3$

The correct values are:

• $f'(2) = -27$
• $f'(3) = -17$
• $f'(4) = -7$
• $f'(5) = 3$

Question 3:

The transformation of $g(x)$ by the translation $T = \binom{2}{2}$.

Apply the translation to $g(x) = 3^x + 3$: $g'(x) = g(x - 2) + 2$ Substitute $g(x) = 3^x + 3$: $g'(x) = 3^{(x - 2)} + 3 + 2$ $g'(x) = 3^{x - 2} + 5$ Thus, the correct answer for question 3 is $g'(x) = 3^{x - 2} + 5$.

Question 4:

Determine the truth of the following statements based on the function $g'(x) = 3^{x - 2} + 5$:

1. $g'(0) = \frac{6}{27}$ is True.

   • $g'(0) = 3^{0 - 2} + 5 = 3^{-2} + 5 = \frac{1}{9} + 5 = \frac{46}{9}$.

2. $g'(1) = \frac{6}{9}$ is True.

   • ( g'(1) = 3^1 Touch