Question

If $f\left(x\right)=x^3+1$, then

(a) the value of f at $x=-1$is $f\left(\_\_\_\right)=\_\_\_$

(b) the value of f at $x=2$is $f\left(\_\_\_\right)=\_\_\_$

(c) the net change in the value of f between $x=-1$and $x=2$is$f\left(\_\_\_\right)-f\left(\_\_\_\right)=\_\_\_$

Short answer

(a) The value of f at $x=-1$is$f(-1)=0$

(b) The value of f at $x=2$is $f\left(2\right)=9$

(c) The net change in the value of f between $x=-1$and $x=2$is $f\left(2\right)-f(-1)=9$

Step-by-step solution

Part a. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part a. Step 2. Analyze the information given.

We are given that $f\left(x\right)=x^3+1$, to find the value at $x=-1$we plug the value of x in the function.

Part a. Step 3. Evaluate the results.

We plug the value of x in the function as shown:

$\begin{array}{l}f\left(x\right)=x^3+1\\ f\left(-1\right)={\left(-1\right)}^3+1\\ =-1+1\\ =0\end{array}$

Part b. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part b. Step 2. Analyze the information given.

We are given that $f\left(x\right)=x^3+1$, to find we value at $x=2$we plug the value of x in the function

Part b. Step 3. Evaluate the results.

We plug the value of x in the function as shown:

$$\begin{gathered} \begin{array}{l}f\left(x\right)=x^3+1\\ f\left(2\right)={\left(2\right)}^3+1\\ =8+1\end{array} \\ =9 \end{gathered}$$

Part c. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$

Part c. Step 2. Analyze the information given

We are given that $f\left(x\right)=x^3+1$, to find net change from $x=-1$to $x=2$we must subtract the values of the functions at these two points.

Part c. Step 3. Evaluate the results.

To find net change from $x=-1$to $x=2$we proceed as shown:

$\text{change}=f\left(2\right)-f(-1)=9-0=9$