Question

Let f(x) = x2 - 9a and g(a) = 3 - x?.(f+g)(7) =- (f - 9)(7) =.· (f9)(7) =• (4) (7) -

Answer 1

Given the functions:

`f(x)=x^2-9x`
`g(x)=3-x^2`

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:

`\begin{gathered} (f+g)=(x^2-9x)+(3-x^2) \\ (f+g)=x^2-9x+3-x^2 \end{gathered}`

Order the like terms together and simplify:

`\begin{gathered} (f+g)=x^2-x^2-9x+3 \\ (f+g)=-9x+3 \end{gathered}`

Substitute the expression with x=7 and solve:

`\begin{gathered} (f+g)(7)=-9x+3 \\ (f+g)(7)=-9\cdot7+3 \\ (f+g)(7)=-60 \end{gathered}`

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:

`(f-g)=(x^2-9x)-(3-x^2)`

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:

`(f-g)=x^2-9x-3+x^2`

Order the like terms and simplify:

`\begin{gathered} (f-g)=x^2+x^2-9x-3 \\ (f-g)=2x^2-9x-3 \end{gathered}`

Substitute the expression with x=7 and solve:

`\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}`

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:

`(fg)=(x^2-9x)(3-x^2)`

Multiply each term of the first parentheses with each term of the second parentheses:

`\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}`

Substitute with x=7 and solve:

`\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}`

The result is (fg)(7)=644

4) (f/g)(7) First, divide both functions:

`((f)/(g))=(x^2-9)/(3-x^2)`
`\begin{gathered} ((f)/(g))=((x-9)x)/(3-x^2) \\ ((f)/(g))=((-1)(x-9)x)/((-1)(3-x^2)) \\ ((f)/(g))=((-x+9)x)/((-3+x^2)) \\ ((f)/(g))=((9-x)x)/((x^2-3)) \\ ((f)/(g))=(9x-x^2)/(x^2-3) \end{gathered}`

Substitute with x=7 and solve:

`\begin{gathered} ((f)/(g))(7)=(9\cdot7-7^2)/(7^2-3) \\ ((f)/(g))(7)=(63-49)/(49-3) \\ ((f)/(g))(7)=(14)/(46) \\ ((f)/(g))(7)=(7)/(23) \end{gathered}`

The result is (f/g)(7)= 7/23