range of f(x)=(x-7)^2
Let's find the real values taken by (x -7)2 using its variation table
⇥Let's study the variations of f(x): first find where its derivative f'(x) = 2·(x -7) is zero
⇥Let's solve in ℝ: 2·(x -7) = 0
⇥We simplify the expression by 2
⇥⇥Let's solve in ℝ: x -7 = 0
⇥⇥Solution found: x = 7
⇥f'(x) is zero at x = 7
⇥Built from usual functions, f'(x) is continuous where it is defined: on each interval where it is never zero, it keeps the sign of its value at any point
⇥f'(0) = -14 < 0: f is decreasing on (-∞, 7)
⇥f'(8) = 2 > 0: f is increasing on (7, +∞)
⇥Let's calculate the values of f at the bounds of the table, or its limits at the infinite or excluded bounds
⇥As x tends to -∞, f(x) tends to +∞
⇥f(7) = (7 -7)2 = 0
⇥As x tends to +∞, f(x) tends to +∞
⇥Let's draw up the variation table:
⇥
x-∞7+∞
f'(x)-0+
f(x)+∞↘0↗+∞
⇥So f is decreasing on (-∞, 7] and increasing on [7, +∞)
On ℝ, f is continuous: it takes every value of [0, +∞)
▶Range:   [0, +∞)