Suppose that the functions f and g are defined for all real numbers x as follows.f(x)=x+2g(x)=3x?Write the expressions for (&f)(x) and (g+f)(x) and evaluate (g-f)(2).(8-1)(x) = 0(8 +5)(x) = 0(8-5)(2) = 0=Х5?

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KeylerO372370 KeylerO372370 · Answer 1 · 2022-10-30T06:37:12-04:00

Given:

[tex]\begin{gathered} f(x)=x+2 \\ g(x)=3x \end{gathered}[/tex]

Required:

[tex]\begin{gathered} (fg)(x)=? \\ (g+f)(x)=? \\ (g-f)(2)=? \end{gathered}[/tex]

Explanation:

The algebraic operations of function f(x) and g(x) is given as

[tex](g+f)(x)=g(x)+f(x)[/tex]

Substituting the value of the functions in the above equation we get

[tex]\begin{gathered} (g+f)(x)=g(x)+f(x) \\ (g+f)(x)=3x+x+2 \\ (g+f)(x)=4x+2 \end{gathered}[/tex]

Now algebraic operation for (f g)(x), we get

[tex](fg)(x)=f(x)g(x)[/tex]

Substituting the values of function we get

[tex]\begin{gathered} (fg)(x)=f(x)g(x) \\ (fg)(x)=(x+2)(3x) \\ (fg)(x)=3x^2+6x \end{gathered}[/tex]

To find (g-f)(2), we first need to find (g-f)(x), which is given by

[tex](g-f)(x)=g(x)-f(x)[/tex]

Substituting the values of function we get

[tex]\begin{gathered} (g-f)(x)=g(x)-f(x) \\ (g-f)(x)=3x-(x+2) \\ (g-f)(x)=3x-x-2 \\ (g-f)(x)=2x-2 \end{gathered}[/tex]

Now lets find (g-f)(2), for this substitute x = 2 in the above equation, we get

[tex]\begin{gathered} (g-f)(x)=2x-2 \\ (g-f)(2)=2(2)-2=4-2 \\ (g-f)(x)=2 \end{gathered}[/tex]

Final answer:

[tex]\begin{gathered} (fg)(x)=3x^2+6x \\ (g+f)(x)=4x+2 \\ (g-f)(2)=2 \end{gathered}[/tex]