If the function g is defined by g (x) = x/(x+1)', which of the following is true?

Rationale

g(10) < g(20)

The function g(x) = x/(x+1) is an increasing function for all x ≥ 0. Therefore, when comparing g(10) and g(20), the output of g(20) will always be greater than that of g(10), confirming that g(10) < g(20).

A (g (10) <g (20)) CORRECT

This statement is true because the function g(x) = x/(x+1) is increasing for all x ≥ 0. As x increases, the value of g(x) also increases, leading to g(20) being greater than g(10).

B (g (20) <g (10)) YOUR ANSWER

This statement is false. As established, g(x) is an increasing function, so g(20) cannot be less than g(10). Instead, g(20) exceeds g(10).

C (g(0) =1) YOUR ANSWER

This statement is incorrect. Evaluating g(0) gives g(0) = 0/(0+1) = 0. Therefore, g(0) does not equal 1; it equals 0.

D (g(1)=0) YOUR ANSWER

This statement is also false. Calculating g(1) results in g(1) = 1/(1+1) = 1/2, which is not equal to 0.

Conclusion

The function g(x) = x/(x+1) is characterized by its increasing nature for non-negative values of x, confirming that g(10) is less than g(20). The incorrect choices either misinterpret the function’s behavior or provide erroneous evaluations at specific points. Understanding the properties of the function is essential for accurately analyzing inequalities and function values.