Find the Domain and Range g(x)=1-2x^2

Problem

g(x)=1−2*x^2

Solution

1. Identify the domain of the function. Since g(x) is a polynomial, it is defined for all real numbers.

D=(−∞,∞)

2. Analyze the structure of the function to find the range. The term x^2 is always greater than or equal to 0 for all real x

x^2≥0

3. Apply transformations to the inequality. Multiply by −2 which reverses the inequality sign.

−2*x^2≤0

4. Add the constant term to both sides of the inequality to isolate g(x)

1−2*x^2≤1

5. Determine the range based on the inequality. Since g(x)≤1 the values of the function span from negative infinity up to and including 1

R=(−∞,1]

Final Answer

Domain: *(−∞,∞), Range: *(−∞,1]

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