Solve For A & B

[mathdad]

Given f(x) = -x^2 + 3 and g(x) = -3x + 3, solve A and B.

A. f(x) > g(x)
B. f(x) = g(x)

Solution for A

f(x) > g(x)

-x^2 + 3 > -3x + 3

-x^2 + 3 + 3x - 3 > 0

-x^2 + 3x > 0

x(-x + 3) > 0

x(3 - x) > 0

<-------0-----------3--------->

Select a number from each region.

Let x = -1

-(-1)^2 + 3(-1) > 0

-1 - 3 > 0

-4 > 0 is false.

Let x = 1

-(1)^2 + 3(1) > 0

-1 + 3 > 0

2 > 0 is true.

Let x = 4

-(4)^2 + 3(4) > 0

-16 + 12 > 0

-4 > 0 is false.

Solution: (0, 3).

Is this right?

Solution for B

f(x) = g(x)

-x^2 + 3 = -3x + 3

Solve for x.

-x^2 + 3x = 3 - 3

-x^2 + 3x = 0

Factor left side.

x(-x + 3) = 0

x = 0

-x + 3 = 0

-x = -3

x = -3/-1

x = 3

Correct?

 

[Dr.Peterson]

Both are correct, but you should make it easier to see that your answer to (B) is not just x = 3, but x = 0 or 3.

 

[mathdad]

Both are correct, but you should make it easier to see that your answer to (B) is not just x = 3, but x = 0 or 3.

Excellent! Another victory for me. Please, check out my new posts in the forum Math, Odds & Ends.