The product of two functions, (fg)(x), is found by multiplying f(x) and g(x). We can write √x as x^(1/2). When multiplying powers with the same base, we add the exponents, so x^(1/2) · x^1 = x³/2.

The product of two functions, (fg)(x), is found by multiplying f(x) and g(x). We can write √x as x^(1/2). When multiplying powers with the same base, we add the exponents, so x^(1/2) · x^1 = x³/2.

Let $f(x)=\sqrt{x}$ 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 $\left(\frac{f}{g}\right)(x)$ .

Answer: $(f+g)(x) = \sqrt{x} + x \\ (f-g)(x) = \sqrt{x} - x \\ (fg)(x) = x^{3/2} \\ \left(\frac{f}{g}\right)(x) = \frac{1}{\sqrt{x}}$

Step-by-step solution

Step 1: Define the functions

We are given two functions, $f(x) = \sqrt{x}$ and $g(x) = x$ . Both functions are defined for nonnegative real numbers, meaning $x \ge 0$ .

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 $\sqrt{x} + x$ .

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 $\sqrt{x} - x$ .

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 $\sqrt{x}$ as $x^{1/2}$ . When multiplying powers with the same base, we add the exponents, so $x^{1/2} \cdot x^1 = x^{3/2}$ .

Step 5: Calculate the quotient of functions

The quotient of two functions, $\left(\frac{f}{g}\right)(x)$ , is found by dividing $f(x)$ by $g(x)$ . We subtract the exponents when dividing powers with the same base, so $x^{1/2} / x^1 = x^{-1/2}$ . This can also be written as $\frac{1}{\sqrt{x}}$ . Note that for the quotient, $g(x)$ cannot be zero, so $x \ne 0$ .

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Let f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=xg(x)=xg(x)=x be two functions defined over the set of nonnegative real numbers. Find (f+g)(x),(f−g)(x),(fg)(x)(f+g)(x),(f-g)(x),(f g)(x)(f+g)(x),(f−g)(x),(fg)(x) and (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x).