Select the correct answer.
Which function has a domain of all real numbers?
A. [tex]$y=(2 x)^{\frac{1}{3}}-7$[/tex]
B. [tex]$y=-2(3 x)^{\frac{1}{6}}$[/tex]
C. [tex]$y=-x^{\frac{1}{2}}+5$[/tex]
D. [tex]$y=(x+2)^{\frac{1}{4}}$[/tex]
Answer (1)
The function with a domain of all real numbers is found by analyzing the exponents and bases of each function.
Function A, y = ( 2 x ) 3 1 − 7 , has an odd root (cube root), allowing all real numbers as its domain.
Functions B, C, and D have even roots, restricting their domains to non-negative numbers or values that make the base non-negative.
Therefore, the correct answer is A .
Explanation
Understanding the Problem We are given four functions and we need to find the one with a domain of all real numbers. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a function with a fractional exponent n 1 , if n is odd, the domain is all real numbers. If n is even, the base must be non-negative.
Analyzing Option A A. y = ( 2 x ) 3 1 − 7 : The exponent is 3 1 , so the base is raised to the cube root. Cube roots are defined for all real numbers. Therefore, the domain is all real numbers.
Analyzing Option B B. y = − 2 ( 3 x ) 6 1 : The exponent is 6 1 , so the base is raised to the sixth root. Sixth roots are only defined for non-negative numbers. Therefore, 3 x ≥ 0 , which means x ≥ 0 . The domain is x ≥ 0 .
Analyzing Option C C. y = − x 2 1 + 5 : The exponent is 2 1 , so the base is raised to the square root. Square roots are only defined for non-negative numbers. Therefore, x ≥ 0 . The domain is x ≥ 0 .
Analyzing Option D D. y = ( x + 2 ) 4 1 : The exponent is 4 1 , so the base is raised to the fourth root. Fourth roots are only defined for non-negative numbers. Therefore, x + 2 ≥ 0 , which means x ≥ − 2 . The domain is x ≥ − 2 .
Conclusion Comparing the domains of the four functions, only function A has a domain of all real numbers. Therefore, the correct answer is A.
Examples
Understanding domains is crucial in many real-world applications. For example, when modeling the growth of a plant, the domain represents the time frame over which the model is valid. Time cannot be negative, so the domain would be restricted to non-negative values. Similarly, in physics, the domain of a function describing the trajectory of a projectile might be limited by physical constraints such as the height of a building or the range of a cannon. Recognizing these limitations ensures that the model provides meaningful and accurate predictions.
