Chapter 1, Problem 23CR

(a)

To determine

To calculate: The value of the function $f\left(x\right)=x^2+1$ .

Answer to Problem 23CR

The value of the function is $10$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=x^2+1$ and the expression $f\left(-3\right)$ .

Formula used:

To find the value of the function, substitute the value in the function and solve.

Calculation:

Consider the provided function $f\left(x\right)=x^2+1$ and the expression $f\left(-3\right)$ .

Recall that to find the value of the function, substitute the value in the function and solve.

So, substitute $x=-3$ in the expression $x^2+1$ and solve it further as,

  $\begin{array}{c}f\left(x\right)=x^2+1\\ f\left(-3\right)={\left(-3\right)}^2+1\\ =9+1\\ =10\end{array}$

Thus, the value of function is $10$ .

(b)

To determine

To calculate: The value of the function $f\left(x\right)=x^2+1$ .

Answer to Problem 23CR

The value of the function is $b^6+1$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=x^2+1$ and the expression $f\left(b^3\right)$ .

Formula used:

To find the value of the function, substitute the value in the function and solve.

Law of exponents ${\left(a^m\right)}^n=a^{mn}$ .

Calculation:

Consider the provided function $f\left(x\right)=x^2+1$ and the expression $f\left(b^3\right)$ .

Recall that to find the value of the function, substitute the value in the function and solve.

So, substitute $x=b^3$ in the expression $x^2+1$ and solve it further as,

  $\begin{array}{c}f\left(x\right)=x^2+1\\ f\left(b^3\right)={\left(b^3\right)}^2+1\\ =b^{3\cdot 2}+1\\ =b^6+1\end{array}$

Thus, the value of function is $b^6+1$ .

(c)

To determine

To calculate: The value of the function $f\left(x\right)=x^2+1$ .

Answer to Problem 23CR

The value of the function is $x^2-2x+2$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=x^2+1$ and the expression $f\left(x-1\right)$ .

Formula used:

To find the value of the function, substitute the value in the function and solve.

The special factoring formula for perfect square ${\left(\text{A}-\text{B}\right)}^2={\text{A}}^2-2\text{AB}+{\text{B}}^2$ .

Calculation:

Consider the provided function $f\left(x\right)=x^2+1$ and the expression $f\left(x-1\right)$ .

Recall that to find the value of the function, substitute the value in the function and solve.

So, substitute $x=x-1$ in the expression $x^2+1$ and solve it further as,

  $\begin{array}{c}f\left(x\right)=x^2+1\\ f\left(x-1\right)={\left(x-1\right)}^2+1\\ =x^2-2\left(x\right)\left(1\right)+1^2+1\\ =x^2-2x+2\end{array}$

Thus, the value of function is $x^2-2x+2$ .