Given the functions:
[tex]f(x)=x^2-9x[/tex][tex]g(x)=3-x^2[/tex]1) (f+g)(x) You have to calculate the sum between f(x) and g(x) for x=7
First, calculate the sum between both functions:
[tex]\begin{gathered} (f+g)=(x^2-9x)+(3-x^2) \\ (f+g)=x^2-9x+3-x^2 \end{gathered}[/tex]Order the like terms together and simplify:
[tex]\begin{gathered} (f+g)=x^2-x^2-9x+3 \\ (f+g)=-9x+3 \end{gathered}[/tex]Substitute the expression with x=7 and solve:
[tex]\begin{gathered} (f+g)(7)=-9x+3 \\ (f+g)(7)=-9\cdot7+3 \\ (f+g)(7)=-60 \end{gathered}[/tex]The result is (f+g)(7)= -60
2) (f-g)(7) You have to calculate the difference between f(x) and g(x) for x=7
First, calculate the difference between both functions:
[tex](f-g)=(x^2-9x)-(3-x^2)[/tex]First, erase the parentheses, the minus sign before (3-x²) indicates that you have to change the sign of both terms inside the parentheses, as if they were multiplied by -1, then:
[tex](f-g)=x^2-9x-3+x^2[/tex]Order the like terms and simplify:
[tex]\begin{gathered} (f-g)=x^2+x^2-9x-3 \\ (f-g)=2x^2-9x-3 \end{gathered}[/tex]Substitute the expression with x=7 and solve:
[tex]\begin{gathered} (f-g)(7)=2x^2-9x+3 \\ (f-g)(7)=2(7)^2-9\cdot7+3 \\ (f-g)(7)=2\cdot49-63-3 \\ (f-g)(7)=98-66 \\ (f-g)(7)=32 \end{gathered}[/tex]The result is (f-g)(7)= 32
3) (fg)(7) In this item you have to calculate the product of f(x) and g(x) for x=7
First, determine the product between both functions:
[tex](fg)=(x^2-9x)(3-x^2)[/tex]Multiply each term of the first parentheses with each term of the second parentheses:
[tex]\begin{gathered} (fg)=x^2\cdot3+x^2\cdot(-x^2)-9x\cdot3-9x\cdot(-x^2) \\ (fg)=3x^2-x^4-27x+9x^3 \\ (fg)=-x^4+9x^3+3x^2-27x \end{gathered}[/tex]Substitute with x=7 and solve:
[tex]\begin{gathered} (fg)(7)=-(7^4)+9\cdot(7^3)+3\cdot(7^2)-27\cdot7 \\ (fg)(7)=-2401+9\cdot343+3\cdot49-189 \\ (fg)(7)=-2401+3087+147-189 \\ (fg)(7)=644 \end{gathered}[/tex]The result is (fg)(7)=644
4) (f/g)(7) First, divide both functions:
[tex](\frac{f}{g})=\frac{x^2-9}{3-x^2}[/tex][tex]\begin{gathered} (\frac{f}{g})=\frac{(x-9)x}{3-x^2} \\ (\frac{f}{g})=\frac{(-1)(x-9)x}{(-1)(3-x^2)} \\ (\frac{f}{g})=\frac{(-x+9)x}{(-3+x^2)} \\ (\frac{f}{g})=\frac{(9-x)x}{(x^2-3)} \\ (\frac{f}{g})=\frac{9x-x^2}{x^2-3} \end{gathered}[/tex]Substitute with x=7 and solve:
[tex]\begin{gathered} (\frac{f}{g})(7)=\frac{9\cdot7-7^2}{7^2-3} \\ (\frac{f}{g})(7)=\frac{63-49}{49-3} \\ (\frac{f}{g})(7)=\frac{14}{46} \\ (\frac{f}{g})(7)=\frac{7}{23} \end{gathered}[/tex]The result is (f/g)(7)= 7/23
