Solution:
The given function is f(x) = log10(1 + x).
We need to find the range of this function for the domain of real values of x when 0 ≤ x ≤ 9.

Range of a function:
The range of a function is the set of all possible output values (y-values) of the function.

Domain of the given function:
The domain of the given function is 0 ≤ x ≤ 9, which means that x can take any real value between 0 and 9, including 0 and 9.

Finding the range of the given function:
To find the range of the given function, we need to analyze the behavior of the function for different values of x.

Case 1: When x = 0
f(0) = log10(1 + 0) = log10(1) = 0
Therefore, the function takes the value 0 when x = 0.

Case 2: When x > 0
As x increases from 0 to 9, the value of 1 + x also increases.
This means that the argument of the logarithmic function log10(1 + x) also increases.
As the argument of the logarithmic function increases, the value of the function also increases.
However, the function value increases at a decreasing rate.
This is because the logarithmic function is an increasing function, but its rate of increase decreases as the argument increases.

Case 3: When x = 9
When x = 9, the argument of the logarithmic function is 1 + 9 = 10.
The value of log10(10) is 1.
Therefore, the function takes the value 1 when x = 9.

Conclusion:
From the above analysis, we can see that the function takes all values between 0 and 1 for 0 < x="" />< />
Therefore, the range of the function for the given domain is {y : 0 < y="" />< />
The correct option is (c) {0.1}.

Function can take any value from 0 to 9 so minimum at x=0 then f(x)=0 maximym value is 9 then f(x)=1 so range is[0,1].