Math

QuestionGraph the function f(x)=12x+1f(x)=1-2 \sqrt{x+1}.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=1x+1f(x)=1- \sqrt{x+1} . The function is defined for all real numbers x such that x1x \geq -1 (since the square root of a negative number is not a real number)
3. We are looking for the graph of this function

STEP 2

We will start by identifying the key features of the function. The function is a transformation of the basic square root function x\sqrt{x}, which is shifted1 unit to the left and1 unit up, and reflected in the x-axis.

STEP 3

The domain of the function is all real numbers x such that x1x \geq -1. This is because the expression under the square root, x+1x+1, must be greater than or equal to0.
x+10x+1 \geq0

STEP 4

olve the inequality x+10x+1 \geq0 for x to find the domain.
x1x \geq -1

STEP 5

The range of the function is all real numbers y such that y1y \leq1. This is because the maximum value of the function is1, which occurs when x=1x = -1.

STEP 6

The graph of the function starts at the point (-1,1) and decreases as x increases. This is because the function is a reflection of the basic square root function in the x-axis.

STEP 7

Plot the graph of the function f(x)=12x+1f(x)=1-2 \sqrt{x+1} using the key features identified.
The graph of the function f(x)=12x+1f(x)=1-2 \sqrt{x+1} is a downward-opening curve that starts at the point (-1,1) and decreases as x increases. The domain of the function is all real numbers x such that x1x \geq -1, and the range is all real numbers y such that y1y \leq1.

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