We are given two real functions, f(x) = x² and g(x) = 2x+1. We need to find their sum, difference, product, and quotient.

Let $f(x)=x^{2}$ and $g(x)=2 x+1$ be two real functions. Find $(f+g)(x),(f-g)(x),(f g)(x),\left(\frac{f}{g}\right)(x) .$

Answer: $(f+g)(x) = x^2 + 2x + 1$ , $(f-g)(x) = x^2 - 2x - 1$ , $(fg)(x) = 2x^3 + x^2$ , $\left(\frac{f}{g}\right)(x) = \frac{x^2}{2x+1}$ for $x \neq -\frac{1}{2}$ .

Step-by-step solution

Step 1: Define the functions

We are given two real functions, $f(x) = x^2$ 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 $x^2$ by each term in $2x+1$ .

Step 5: Calculate the quotient of functions

The quotient of two functions, $\left(\frac{f}{g}\right)(x)$ , 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 $x \neq -\frac{1}{2}$ .

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Let f(x)=x2f(x)=x^{2}f(x)=x2 and g(x)=2x+1g(x)=2 x+1g(x)=2x+1 be two real functions. Find (f+g)(x),(f−g)(x),(fg)(x),(fg)(x).(f+g)(x),(f-g)(x),(f g)(x),\left(\frac{f}{g}\right)(x) .(f+g)(x),(f−g)(x),(fg)(x),(gf​)(x).