Question

F(x) = -x² + 5 and g(x) = -2x+2, find all values of x for which f(x) = g(x).

Answer 1

Equate Both equatios and find x

• f(x)=g(x)
• -x²+5=-2x+2
• x²-5=2x-2
• x²-2x-3=0
• (x+1)(x-3)=0

The points are -1,3

Answer 2

The values of x for which f(x) = g(x) are:

• x = -1
• x = 3

Explanation:

To find all the values of x for which f(x) = g(x), equate the given functions:

`\implies f(x)=g(x)`

`\implies -x^2+5=-2x+2`

Rearrange the equation so that it is in the quadratic form ax²+bx+c=0:

`\implies -x^2+5+2x-2=0`

`\implies -x^2+2x+3=0`

Divide both sides by -1:

`\implies x^2-2x-3=0`

Factor the quadratic equation:

`\implies x^2-3x+x-3=0`

`\implies x(x-3)+1(x-3)=0`

`\implies (x+1)(x-3)=0`

To solve for x, apply the zero-product property:

`(x+1)=0 \implies x=-1`

`(x-3)=0 \implies x=3`

Therefore, the values of x for which f(x) = g(x) are:

• x = -1
• x = 3

We can check the solutions by substituting each found value of x into the functions.

`f(-1)=-(-1)^2+5=4`

`g(-1)=-2(-1)+2=4`

`f(3)=-(3)^2+5=-4`

`g(3)=-2(3)+2=-4`

As f(-1) = g(-1) and f(3) = g(3), this proves that the values of x for which f(x) = g(x) are x = -1 or x = 3.