Chapter 2, Problem 13RE

To determine

To find: The pair of function from the provided functions.

Answer to Problem 13RE

Solution:

a. The domain of the pair $f+g$ is the set of all real numbers.

b. The domain of the pair $f-g$ is the set of all real numbers.

c. The domain of the pair $f\circ g$ is the set of all real numbers.

d. The domain of the pair $\frac{f}{G}$ is $\left\{x\right|x\ne 0,x\in ℝ\}$.

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=3x^2+x+1$ and $g\left(x\right)=3x$.

Formula used:

A quadratic equation is any equation/function with a degree of 2 that can be written in the form $y=a{x}^2+bx+c$, where $a,b$ and $c$ are real numbers, and a does not equal to 0. Its graph is called a parabola. The constants $a,b$, and $c$ are called the parameters of the equation. The values of $a,b$ and $c$ determine the shape and position of the parabola.

The domain of a function is the set of all real values of $x$ that will give real values for $y$. The range of a function is the set of all real values of $y$ that you can get by plugging real numbers into $x$.

Calculation:

a. For $f+g$, $f\left(x\right)+g\left(x\right)=3x^2+x+1+3x=3x^2+4x+1$. Here the function $f+g$ is a quadratic equation. By the definition of domain of the quadratic equation, the domain takes all the real values. Therefore, the domain of the pair $f+g$ is the set of all real numbers.

b. For $f-g$, $f\left(x\right)-g\left(x\right)=3x^2+x+1-3x=3x^2-2x+1$. Here the function $f-g$ is a quadratic equation. By the definition of domain of the quadratic equation, the domain takes all the real values. Therefore, the domain of the pair $f-g$ is the set of all real numbers.

c. For $f\circ g$, $f\left(x\right)\cdot g\left(x\right)=\left(3x^2+x+1\right)\times \left(3x\right)=9x^3+3x^2+3x$. Here the function $f\circ g$ is an algebraic equation of degree 3. Therefore, there will not be any restriction for this equation. The input takes all the real values. Therefore, the domain of the pair $f\circ g$ is the set of all real numbers.

d. For $\frac{f}{G}$, $\frac{f\left(x\right)}{g\left(x\right)}=\frac{3x^2+x+1}{3x}$. The function $\frac{f}{G}$ says to divide $3x^2+x+1$ by $3x$. Since division by 0 is not defined, the denominator $3x$ can never be 0, so $x$ can never equal 0. The domain of the pair $\frac{f}{G}$ is $\left\{x\right|x\ne 0,x\in ℝ\}$.