Chapter 2.1, Problem 44AYU

In problems 43-50, find the following for each function:

(a) $f\left(0\right)$ (b) $f\left(1\right)$ (c) $f(-1)$ (d) $f(-x)$ (e) $-f\left(x\right)$ (f) $f\left(x\text{ + }1\right)$ (g) $f\left(2x\right)$ (h) $f\left(x+h\right)$

$f\left(x\right)=\sqrt{x^2+}x$

To determine

To find: $f\left(0\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(0\right)=\text{ 0}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(0)=\sqrt{0^2+0}=0$

To determine

To find: $f\left(1\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(1\right)=\sqrt{2}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(1)=\sqrt{1^2+1}=\sqrt{2}$

To determine

To find: $f\left(-1\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

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

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(-1)=\sqrt{{(-1)}^2+\left(-1\right)}=\sqrt{1-1}=0$

To determine

To find: $f\left(-x\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(-x\right)=\sqrt{x^2-x}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(-x)=\sqrt{{(-x)}^2+\left(-x\right)}=\sqrt{x^2-x}$

To determine

To find: $-f\left(x\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$-f\left(x\right)=-\sqrt{x^2+x}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$-f(x)=-\sqrt{x^2+x}$

To determine

To find: $f\left(x+1\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(x+1\right)=\sqrt{x^2+3x+2}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(x+1)=\sqrt{{(x+1)}^2+\left(x+1\right)}$

$=\sqrt{x^2+2x+1+x+1}$

$=\sqrt{x^2+3x+2}$

To determine

To find: $f\left(2x\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(2x\right)=\sqrt{4x^2+2x}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(2x)=\sqrt{{(2x)}^2+2x}$

$=\sqrt{4x^2+2x}$

To determine

To find: $f\left(x+h\right)$, for the function $f(x)=\sqrt{x^2+x}$

Answer to Problem 44AYU

$f\left(x+h\right)=\sqrt{x^2+2xh+h^2+x+h}$

Explanation of Solution

Given:

The function $f(x)=\sqrt{x^2+x}$.

Calculation:

$f(x+h)=\sqrt{{(x+h)}^2+\left(x+h\right)}$

$=\sqrt{x^2+2xh+h^2+x+h}$