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Let f(x) be an odd function defined for all real numbers x, and let g(x)=f(x+3)-5. You are told that the graph of y=g(x) passes through the point (2, -2). Then the graph of y=g(x) must also pass through two other points (a,b) and (c,d) Enter your answer in the form "(a,b),(c,d)".

 May 6, 2024
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We can solve this problem by using the properties of odd functions and the given information about f(x) and g(x).

 

Odd Function Property: An odd function satisfies f(-x) = -f(x) for all real numbers x. This means the function is symmetrical about the origin (0, 0).

 

Shifting the Graph: The function g(x) is defined as f(x + 3) - 5. This means the graph of g(x) is obtained by shifting the graph of f(x) three units to the left and five units down.

 

Point on the Graph of g(x): We are given that the graph of y = g(x) passes through the point (2, -2). This translates to f(2 + 3) - 5 = -2, which means f(5) = 3.

 

Odd Function and f(5): Since f(x) is odd, we know f(-5) = -f(5) = -3.

 

Points on the Original Graph (f(x)): Because g(x) is obtained by shifting f(x), the corresponding points on the graph of f(x) are:

 

Point for f(5): (-5, -3) (three units to the left of (2, -2) due to the shift)

 

Point for f(-5): (5, 3) (three units to the right of (2, -2) due to the shift)

 

Points on the Shifted Graph (g(x)): Since g(x) shifts the graph of f(x), the corresponding points on the graph of g(x) are:

 

Point for g(-5): (-2, -3) (three units to the left and five units down from (-5, -3))

 

Point for g(5): (8, 3) (three units to the right and five units down from (5, 3))

 

Therefore, the graph of y = g(x) must also pass through the points:

 

(-2, -3)

(8, 3)

 

So the answer is: (-2, -3), (8, 3)

 May 6, 2024

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