Let f(x)=x2 and g(x)=2x+1 be two real functions. Find (f+g)(x),(f−g)(x),(fg)(x),(gf)(x). $
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Step-by-Step Solution
Step 1: Define the functions
We are given two real functions, f(x)=x2 and g(x)=2x+1. We need to find their sum, difference, product, and quotient.
Step 2: Calculate the sum of functions
The sum of two functions, (f+g)(x), is found by adding their individual expressions. We substitute f(x) and g(x) and combine like terms.
Step 3: Calculate the difference of functions
The difference of two functions, (f-g)(x), is found by subtracting the second function from the first. Remember to distribute the negative sign to all terms of g(x).
Step 4: Calculate the product of functions
The product of two functions, (fg)(x), is found by multiplying their individual expressions. We use the distributive property to multiply x2 by each term in 2x+1.
Step 5: Calculate the quotient of functions
The quotient of two functions, (gf)(x), is found by dividing the first function by the second. It is important to note that the denominator cannot be zero, so we must exclude values of x for which 2x+1=0. This means x=−21.