Use the definitions for the function operations to find the rules of correspondence of the given functions.
[tex]\begin{gathered} f(x)=4x^2+2x+7 \\ g(x)=2x-3 \end{gathered}[/tex]1)
Remember that:
[tex](g-f)(x)=g(x)-f(x)[/tex]Replace the expressions for g(x) and f(x) and simplify:
[tex]\begin{gathered} (g-f)(x)=(2x-3)-(4x^2+2x+7) \\ =2x-3-4x^2-2x-7 \\ =-4x^2-10 \end{gathered}[/tex]2)
Remember that:
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)},g(x)\ne0[/tex]Replace the expressions for f(x)i and g(x):
[tex](\frac{f}{g})(x)=\frac{4x^2+2x+7}{2x-3}[/tex]The domain is the set of all real number such that g(x) is different form 0:
[tex]\begin{gathered} g(x)\ne0 \\ \Rightarrow2x-3\ne0 \\ \Rightarrow2x\ne3 \\ \therefore x\ne\frac{3}{2} \end{gathered}[/tex]Using interval notation, the domain is:
[tex](-\infty,\frac{3}{2})\cup(\frac{3}{2},\infty)[/tex]3)
Remember that:
[tex](f\cdot g)(x)=f(x)\cdot g(x)[/tex]Then:
[tex]\begin{gathered} (f\cdot g)(x)=(4x^2+2x+7)(2x-3) \\ =(4x^2)(2x)+(2x)(2x)+(7)(2x)+(4x^2)(-3)+(2x)(-3)+(7)(-3) \\ =8x^3+4x^2+14x-12x^2-6x-21 \\ =8x^3-8x^2+8x-21 \end{gathered}[/tex]4)
To find f(x-3), replace (x-3) for x in the rule of correspondence of f:
[tex]\begin{gathered} f(x)=4x^2+2x+7 \\ \Rightarrow f(x-3)=4(x-3)^2+2(x-3)+7 \\ =4(x^2-6x+9)+2x-6+7 \\ =4x^2-24x+36+2x+1 \\ =4x^2-22x+37 \end{gathered}[/tex]5)
Remember that:
[tex](f\circ g)=f(g(x))[/tex]To find f(g(x)), replace g(x) for x into the rule of correspondence of f:
[tex]\begin{gathered} (f\circ g)(x)=f(g(x)) \\ =4(g(x))^2+2(g(x))+7 \end{gathered}[/tex]Replace the expression for g(x):
[tex]\begin{gathered} \Rightarrow(f\circ g)(x)=4(2x-3)^2+2(2x-3)+7 \\ =4(4x^2+-12x+9)+4x-6+7 \\ =16x^2-48x+36+4x+1 \\ =16x^2-44x+37 \end{gathered}[/tex]6)
To find g(f(x)), replace f(x) for x into the rule of correspondence of g(x):
[tex]\begin{gathered} (g\circ f)(x)=g(f(x)) \\ =2\cdot f(x)-3 \\ =2(4x^2+2x+7)-3 \\ =8x^2+4x+14-3 \\ =8x^2+4x+11 \end{gathered}[/tex]7)
Notice that we already have a rule of correspondence for g(f(x)). Substitute x=-1 to find g(f(-1)):
[tex]\begin{gathered} g(f(x))=8x^2+4x+11 \\ \Rightarrow g(f(-1))=8(-1)^2+4(-1)+11 \\ =8-4+11 \\ =15 \end{gathered}[/tex]8)
To find the inverse of f(x), repace y=f(x) and isolate x:
[tex]\begin{gathered} y=4x^2+2x+7 \\ \Rightarrow4x^2+2x+7-y=0 \\ \Rightarrow x=\frac{-2+\sqrt[]{2^2-4(4)(7-y)}}{2(4)} \\ =\frac{-2+\sqrt[]{4-112+16y}}{8} \\ =\frac{-2+\sqrt[]{16y-108}}{8} \\ =\frac{-1+\sqrt[]{4y-27}}{4} \end{gathered}[/tex]Next, switch x and y in the equation:
[tex]y=\frac{-1+\sqrt[]{4x-27}}{4}[/tex]Finally, substitute y=f^-1(x):
[tex]\therefore f^{-1}(x)=\frac{-1+\sqrt[]{4x-27}}{4}[/tex]9)
To find f(-x), replace x for -x in the rule of correspondence of f:
[tex]\begin{gathered} f(-x)=4(-x)^2+2(-x)+7 \\ =4x^2-2x+7 \end{gathered}[/tex]