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QuestionWhich function has a restricted domain? a. f(x)=x−1f(x)=\sqrt{x}-1 b. f(x)=−x3+4xf(x)=-x^{3}+4 x c. f(x)=−∣x−5∣+1f(x)=-|x-5|+1 d. f(x)=12x+63f(x)=\frac{1}{2} \sqrt[3]{x+6}

Studdy Solution

STEP 1

Assumptions1. We have four functions and we need to determine which one does not have a domain of all real numbers. . The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 2

For each function, we will determine the domain.
a. For f(x)=x−1f(x)=\sqrt{x}-1, the domain is all xx such that x≥0x \geq0. This is because the square root of a negative number is not a real number.
b. For f(x)=−x+4xf(x)=-x^{}+4 x, the domain is all real numbers, because we can cube any real number and then multiply it by a real number.
c. For f(x)=−∣x−5∣+1f(x)=-|x-5|+1, the domain is all real numbers, because the absolute value function is defined for all real numbers.
d. For f(x)=12x+6f(x)=\frac{1}{2} \sqrt[]{x+6}, the domain is all real numbers, because the cube root function is defined for all real numbers.

STEP 3

From the above analysis, we can see that the function f(x)=x−1f(x)=\sqrt{x}-1 does not have a domain of all real numbers.So, the answer is option a f(x)=x−1f(x)=\sqrt{x}-1.

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