Question

Find and compare the domain and range of each function. f(x) = 5/8x + 1 g(x) = 2/3x^3 A. The domains and ranges of both functions are not restricted. B. The domain of f is restricted, but its range is not. The domain and range of g are not restricted. C. The range of f is restricted, but its domain is not. The domain and range of g are not restricted. D. Both the domain and range of f are restricted, but the domain and range of g are not.

Answer 1

A. The domains and ranges of both functions are not restricted.

For f(x) = 5/8x + 1, any real number can be plugged in for x, so the domain is (-∞, ∞). Similarly, any real number can be obtained by evaluating the function, so the range is also (-∞, ∞).

For g(x) = 2/3x^3, any real number can be plugged in for x, so the domain is (-∞, ∞). As x is cubed, the range can be negative or positive infinity, or any real number in between, so the range is also (-∞, ∞).

Therefore, both functions have unrestricted domains and ranges.

The correct answer is A.

Answer 2

Both functions f(x) and g(x) have unrestricted domains and ranges, meaning they can take all real numbers as input (domain) and can produce all real numbers as output (range).

Step-by-step explanation:

The functions f(x) = \frac{5}{8}x + 1 and g(x) = \frac{2}{3}x^3 have different characteristics for their domains and ranges. For the linear function f(x), the domain consists of all real numbers since there is no restriction on the possible values of x, and its range is also all real numbers because as x takes any real value, the output can be any real number as well. On the other hand, the cubic function g(x) also has a domain of all real numbers because there is no restriction on the values that x can take. Its range is also all real numbers since raising any real number to the third power and multiplying it by a constant can yield any real number as a result.

The correct answer comparing the domain and range of these functions is option A. The domains and ranges of both functions are not restricted.