The functions f and g are defined as follows. f(x) = -x + 2
g(x) = x^2 - 1 The point (b,c) is on both the graph of f and the graph of g in the xy-plane. Which of the following inequalities must be true?
To determine which of the given inequalities must be true, we need to find the values of b and c that satisfy both equations f(x) = -x + 2 and g(x) = x^2 - 1.
First, let's find the values of x where f(x) = g(x). We can set the two equations equal to each other:
-x + 2 = x^2 - 1
Rearranging the equation gives us:
x^2 + x - 3 = 0
Now, we can solve this quadratic equation by factoring or by using the quadratic formula. Factoring the equation, we have:
(x + 3)(x - 1) = 0
This gives us two potential solutions for x: x = -3 and x = 1.
Now, let's substitute these values back into the original equations to find the corresponding points (b, c):
For x = -3:
f(-3) = -(-3) + 2 = 3 + 2 = 5
g(-3) = (-3)^2 - 1 = 9 - 1 = 8
So, when x = -3, the corresponding point is (-3, 5) for f(x) and (-3, 8) for g(x).
For x = 1:
f(1) = -(1) + 2 = -1 + 2 = 1
g(1) = (1)^2 - 1 = 1 - 1 = 0
So, when x = 1, the corresponding point is (1, 1) for f(x) and (1, 0) for g(x).
Now, we can determine which of the given inequalities hold true for these points.
The possible inequalities are not provided in the question, so you will need to provide the options for me to evaluate and determine which one(s) must be true based on the given points (b,c).
To find the point (b, c) that is on both the graphs of f and g, we need to equate the two functions: f(x) = g(x)
Given:
f(x) = -x + 2
g(x) = x^2 - 1
Equating the functions:
-x + 2 = x^2 - 1
Rearranging the equation:
x^2 + x - 3 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula:
Factoring:
(x + 3)(x - 1) = 0
By setting each factor equal to zero, we get two possible solutions:
x + 3 = 0 --> x = -3
x - 1 = 0 --> x = 1
Now we substitute each value of x back into either f(x) or g(x) to find the corresponding y-coordinate (c):
When x = -3:
f(-3) = -(-3) + 2 = 5
g(-3) = (-3)^2 - 1 = 8
When x = 1:
f(1) = -(1) + 2 = 1
g(1) = (1)^2 - 1 = 0
So, the two points (b, c) that are on both the graphs of f and g are (-3, 5) and (1, 0).
Now let's check which of the given inequalities must be true for these points:
1) b < c: (-3 < 5), which is true.
2) b > c: (-3 > 5), which is false.
3) c < b: (5 < -3), which is false.
4) c > b: (5 > -3), which is true.
Based on these calculations, the only inequality that must be true is: c > b.