Let f(x)=x and g(x)=x be two functions defined over the set of nonnegative real numbers. Find (f+g)(x),(f-g)(x),(f g)(x) and (gf)(x).
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Step-by-Step Solution
Step 1: Define the functions
We are given two functions, f(x)=x and g(x)=x. Both functions are defined for nonnegative real numbers, meaning x≥0.
Step 2: Calculate the sum of functions
The sum of two functions, (f+g)(x), is found by adding their individual expressions. So, we add f(x) and g(x) to get x+x.
Step 3: Calculate the difference of functions
The difference of two functions, (f-g)(x), is found by subtracting g(x) from f(x). This results in x−x.
Step 4: Calculate the product of functions
The product of two functions, (fg)(x), is found by multiplying f(x) and g(x). We can write x as x1/2. When multiplying powers with the same base, we add the exponents, so x1/2⋅x1=x3/2.
Step 5: Calculate the quotient of functions
The quotient of two functions, (gf)(x), is found by dividing f(x) by g(x). We subtract the exponents when dividing powers with the same base, so x1/2/x1=x−1/2. This can also be written as x1. Note that for the quotient, g(x) cannot be zero, so x
e 0.