Math

Question Given f(x)=x+2+3f(x)=\sqrt{x+2}+3, choose the correct transformation: left 2, up 3.

Studdy Solution

STEP 1

Assumptions
1. We are given the function f(x)=x+2+3f(x) = \sqrt{x+2} + 3.
2. We need to determine the correct transformation of the function compared to the base function g(x)=xg(x) = \sqrt{x}.
3. The transformations can be horizontal (left or right) and vertical (up or down).

STEP 2

Identify the base function and the transformations applied to it to obtain the given function.
The base function is g(x)=xg(x) = \sqrt{x}.
The given function is f(x)=x+2+3f(x) = \sqrt{x+2} + 3.

STEP 3

Analyze the horizontal transformation.
The term under the square root in f(x)f(x) is x+2x+2, which indicates a horizontal shift. Since we are adding 2 inside the square root, this corresponds to a shift to the left by 2 units.

STEP 4

Analyze the vertical transformation.
The term outside the square root in f(x)f(x) is +3+3, which indicates a vertical shift. Since we are adding 3 outside the square root, this corresponds to a shift upwards by 3 units.

STEP 5

Combine the transformations.
The function f(x)=x+2+3f(x) = \sqrt{x+2} + 3 is transformed from the base function g(x)=xg(x) = \sqrt{x} by shifting left by 2 units and up by 3 units.

STEP 6

Choose the correct transformation from the given options.
The correct transformation is "left 2, up 3".

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