Question

Which function would have a RANGE of all real numbers?

a) f(x) = x^2 - 8
b) f(x) = sqrt{x} - 8 )
c) f(x) = -8
d) f(x) = x - 8

Answer 1

The function with a range of all real numbers is d) f(x) = x - 8, which is a linear function and its range includes all real numbers as for any real value of x, f(x) also takes on a real value.

Step-by-step explanation:

The question is asking which function has a range of all real numbers. Let's examine each option:

• f(x) = x^2 - 8 is a quadratic function. Its graph is a parabola opening upwards, which means that the range is limited to values greater than or equal to the vertex's y-coordinate (-8 in this case).
• f(x) = sqrt{x} - 8 is a square root function translated 8 units down. Its range starts from -8 and goes upwards, so not all real numbers are included.
• f(x) = -8 is a constant function. The range is just the single value -8.
• f(x) = x - 8 is a linear function with a slope of 1. As x takes on any real number, so does f(x). Therefore, its range includes all real numbers.

Therefore, the correct answer is d) f(x) = x - 8, because for every real number value of x, there is a corresponding value of f(x), hence its range covers all real numbers.