Math Problem Statement

Suppose a function f (x) is defined by the given coordinates in the table below. Determine the values A, B, C, and D in the table for the function g (x) = 2f (x) +5.

Solution

To solve this problem, we need to use the given function g(x)=2f(x)+5g(x) = 2f(x) + 5 and apply it to each coordinate point of f(x)f(x).

Let’s assume that the coordinates of f(x)f(x) are provided in a table like this:

x & f(x) \\ \hline x_1 & f(x_1) \\ x_2 & f(x_2) \\ x_3 & f(x_3) \\ x_4 & f(x_4) \\ \end{array}$$ Then, the corresponding table for $$g(x) = 2f(x) + 5$$ will be: $$\begin{array}{c|c} x & g(x) \\ \hline x_1 & 2f(x_1) + 5 \\ x_2 & 2f(x_2) + 5 \\ x_3 & 2f(x_3) + 5 \\ x_4 & 2f(x_4) + 5 \\ \end{array}$$ Given that $$g(x)$$ can be represented as $$g(x) = 2f(x) + 5$$, we compute the values $$A$$, $$B$$, $$C$$, and $$D$$ as follows: - $$A = 2f(x_1) + 5$$ - $$B = 2f(x_2) + 5$$ - $$C = 2f(x_3) + 5$$ - $$D = 2f(x_4) + 5$$ ### Steps to determine the values $$A$$, $$B$$, $$C$$, and $$D$$: 1. Multiply each value of $$f(x)$$ by 2. 2. Add 5 to each of the results obtained in step 1. If the actual coordinates $$f(x_1), f(x_2), f(x_3), f(x_4)$$ are known, you can directly compute $$A$$, $$B$$, $$C$$, and $$D$$. Would you like to provide the specific values from the table, or do you need further details on any step? --- Here are 5 related questions to explore: 1. What happens to the graph of $$f(x)$$ when transformed into $$g(x) = 2f(x) + 5$$? 2. How would the function $$g(x)$$ change if the transformation was $$g(x) = 3f(x) - 4$$? 3. Can you determine $$g(x)$$ given different function values for $$f(x)$$? 4. What effect does multiplying a function by a scalar have on its graph? 5. How does adding a constant to a function affect its graph? **Tip:** Understanding how transformations affect functions graphically can help in quickly predicting the outcome of such changes.

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Math Problem Analysis

Mathematical Concepts

Function Transformation
Linear Functions

Formulas

g(x) = 2f(x) + 5

Theorems

Linear transformation theorem

Suitable Grade Level

Grades 9-11