Answer:
Explanation:
Here Is a Quick Explanation :)
To find the axis of symmetry and the vertex of the function, we can use the fact that the function is recursive, and we can write:
f(x) = f(x-1)² + 7
f(x-1) = f(x-2)² + 7
f(x-2) = f(x-3)² + 7
...
f(1) = f(0)² + 7
If we substitute the last equation into the previous one, we get:
f(x-1) = (f(0)² + 7)² + 7
f(x) = ((f(0)² + 7)² + 7)² + 7
This shows that the function depends only on the initial value f(0), and we can use this fact to find the axis of symmetry and the vertex.
To find the axis of symmetry, we need to find the value of x that makes f(x) equal to f(-x). We can write:
f(-x) = f(-(x-1))² + 7 = f(-x+1)² + 7
Now, if we substitute f(x) = f(x-1)² + 7 into the last equation, we get:
f(-x) = (f(x-1)² + 7)² + 7 = f(x)² + 7
This means that the axis of symmetry is the line x = 0, and not y = 7 as stated in option A.
To find the vertex, we need to find the maximum or minimum value of the function. Since f(x) = f(x-1)² + 7, the function is increasing if f(x-1) > -7, and decreasing if f(x-1) < -7. Since f(0) = 7, we can conclude that the function is increasing on the interval -∞ < x < 1, and decreasing on the interval x > 1. Therefore, the vertex is at x = 1, and the corresponding value is f(1) = 7.
Therefore, the correct statements are:
The axis of symmetry of f(x) is x = 0.
The vertex of the function is (1, 7).
f is increasing on the interval -∞ < x < 1.
f is decreasing on the interval x > 1.