Find the set of values of the function y = (sinx + cosx) ^ 2

1. Let us square the sum of trigonometric functions and use the formula for the sine of a double angle:

sin2α = 2sinα * cosα;
y = (sinx + cosx) ^ 2;
y = sin ^ 2x + 2sinx * cosx + cos ^ 2x;
y = 1 + sin2x.
2. If there are no restrictions on the function argument, then the sine takes values in the range from -1 to 1:

sin2x ∈ [-1; 1], or as a double inequality:
-1 ≤ sin2x ≤ 1. (1)
3. Adding unity to all parts of inequality (1), we find the range of values of the original function:

1 – 1 ≤ 1 + sin2x ≤ 1 + 1;
0 ≤ y ≤ 2;
y ∈ [0; 2].
Answer: [0; 2].