Chapter 2, Problem 9CT

For the functions $f(x)=2x^2+1$ and $g(x)=3x-2,$ find the following and simplify.

$(f-g)(x)$

$(f*g)(x)$

$f(x+h)-f(x)$

To determine

The function $(f-g)(x)$, and simplify where $f(x)=2x^2+1$ and $g(x)=3x-2$.

Answer to Problem 9CT

Solution:

The function $(f-g)(x)=2x^2-3x+3$.

Explanation of Solution

Given Information:

The functions are $f(x)=2x^2+1$ and $g(x)=3x-2$.

Explanation:

As $(f-g)(x)=f(x)-g(x)$,

Substitute $f(x)=2x^2+1$ and $g(x)=3x-2$ in above equation,

$\Rightarrow (f-g)(x)=2x^2+1-(3x-2)$

Simplify above equation,

$\Rightarrow (f-g)(x)=2x^2+1-3x+2$

Combine like terms,

$\Rightarrow (f-g)(x)=2x^2-3x+3$

Therefore, the function is $(f-g)(x)=2x^2-3x+3$.

To determine

The function $(f\cdot g)(x)$, and simplify where $f(x)=2x^2+1$ and $g(x)=3x-2$.

Answer to Problem 9CT

Solution:

The function $(f\cdot g)(x)=6x^3-4x^2+3x-2$.

Explanation of Solution

Given Information:

The functions are $f(x)=2x^2+1$ and $g(x)=3x-2$.

Explanation:

As $(f\cdot g)(x)=f(x)\cdot g(x)$,

Substitute $f(x)=2x^2+1$ and $g(x)=3x-2$ in above equation,

$\Rightarrow (f\cdot g)(x)=(2x^2+1)\cdot (3x-2)$.

Simplify above equation,

$\Rightarrow (f\cdot g)(x)=2x^2\cdot 3x-2x^2\cdot 2+1\cdot 3x-1\cdot 2$.

Simplify above equation,

$\Rightarrow (f\cdot g)(x)=6x^3-4x^2+3x-2$.

Therefore, $(f\cdot g)(x)=6x^3-4x^2+3x-2$.

To determine

$f(x+h)-f(x)$, and simplify where $f(x)=2x^2+1$ and $g(x)=3x-2$.

Answer to Problem 9CT

Solution:

The difference $f(x+h)-f(x)=4xh+2h^2$.

Explanation of Solution

Given Information:

The functions are $f(x)=2x^2+1$ and $g(x)=3x-2$.

Explanation:

To find $f(x+h)$, replace $x$ by $x+h$ in $f(x)=2x^2+1$,

$\Rightarrow f(x+h)=2{(x+h)}^2+1$.

Simplify above equation,

$\Rightarrow f(x+h)=2(x^2+2xh+h^2)+1$.

$\Rightarrow f(x+h)=2x^2+4xh+2h^2+1$.

Consider $f(x+h)-f(x)$.

Substitute the value of $f(x+h)$ and $f(x)$ in above expression,

$\Rightarrow f(x+h)-f(x)=2x^2+4xh+2h^2+1-2x^2-1$

Combine like terms of above equation,

$\Rightarrow f(x+h)-f(x)=4xh+2h^2$

Therefore, $f(x+h)-f(x)=4xh+2h^2$.