Question

Suppose that the functions f and g are defined for all real numbers x as follows.f(x) = x + 2g(x) = 3x + 2Write the expressions for (f-g)(x) and (fg)(x) and evaluate (f+g)(-1).

Answer 1

Answer:

(f-g)(x)= -2x

(f.g)(x)= 3x^2 + 8x +4

(f+g)(-1)=0

Explanation:

Given:

f(x) = x + 2

g(x) = 3x + 2

(f-g)(x)=?

(f.g)(x)=?

(f+g)(-1)=?

The expression for (f-g)(x) is:

(f(x)-g(x))(x) = (x + 2) - (3x + 2)

=x+2-3x-2

Simplify

(f(x)-g(x))(x) = -2x

The expression for (f.g)(x) is:

$\begin{gathered} (f(x)\cdot g(x))(x)=(x+2)(3x+2)_{} \\ We\text{ use FOIL method:} \\ (a+b)(c+d)=ac+ad+bc+bd \\ Let\text{: } \\ a=x \\ b=2 \\ c=3x \\ d=2 \\ So, \\ =(x)(3x)+(x)(2)+(2)(3x)+(2)(2) \\ \text{Simplify} \\ =3x^2+2x+6x+4 \\ (f(x)\cdot g(x))(x)=3x^2+8x+4 \end{gathered}$

And, the evaluation for (f+g)(-1) is:

(f(x) +g(x))(-1) = ((x + 2)+(3x + 2))(-1)

This means we let x= -1.

We plug in what we know:

=x + 2+3x + 2)

=-1+2+3(-1)+2

Simplify:

=-1+2-3+2

=-4+4

(f(x) +g(x))(-1) = 0

Therefore:

$\begin{gathered} \mleft(f\mleft(x\mright)-g\mleft(x\mright)\mright)\mleft(x\mright)=-2x \\ (f(x)\cdot g(x))(x)=3x^2+8x+4 \\ \mleft(f\mleft(x\mright)+g\mleft(x\mright)\mright)\mleft(-1\mright)=0 \end{gathered}$