Question

How do you do this problem?

Answer 1

f(x) = (x - 4)^2 - 1

g(x) = -(1/4) ( x - 4)^2 + 4

both the x and y values have to be the same. Start with the y values

f(x) = g(x)

(x - 4)^2 - 1 = -(1/4) (x - 4)^2 + 4 Add 1 to both sides.

(x - 4)^2 = -(1/4) (x - 4)^2 + 5 Add 1/4(x - 4)^2 to both sides.

(5/4) (x - 4)^2 = 5 Divide by 5/4 on both sides.

(x - 4)^2 = 5//(5/4)

(x - 4)^2 = (5/1)//5/4 Invert the second fraction and multiply

(x - 4)^2 = 5/1 * 4/5

(x - 4)^2 = 4 The 5s cancel

(x - 4)^2 = 4 Take the square root of both sides.

(x - 4) = +/- 2 Add 4 to each answer. Start with +2 on the right.

x - 4 + 4 = 2 + 4

x = 4 + 2 = 6

The x value that makes f(x)- g(x) = 0 is x = 6 The point is (6,3) answer.

Answer C.

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You do not need this next part. It is just for completeness.

x - 4 = - 2

x = 4 -2

x = 2

What are the y values for these 2 x values?

y = (x - 4)^2 - 1

y = (6 - 4)^2 - 1

y = 4 - 1

y = 3

The point where f(x) - g(x) = 0 is (6,3) <<<<<< Answer 1

The second point is

y = (x - 4)^2 - 1

y = (2 - 4)^2 - 1

y = (-2)^2 - 1

y = 4 - 1

y = 3

The second point is (2,3). Answer 2

Note the y values are the same. You might expect that.

Answer 2

You can let q = (x-4)². Then you have

... f(q) = q - 1

... g(q) = -q/4 + 4

Putting these into the equation f - g = 0 gives you

...q - 1 - (-q/4 + 4) = 0

... (5/4)q = 5 . . . . collect terms, add 5

... q = (4/5)·5 = 4 . . . . multiply by the inverse of the q coefficient

Relating this back to x, you have

... (x -4)² = 4

... x - 4 = ±2 . . . . take the square root

... x = 4 ± 2 = {2, 6} . . . . . . add 4 to find x (there are two solutions)

The appropriate choice is

... (C) 6