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    Question

    Let f:R→R be defined as f(x) = cosx + sinx. Then the

    range of f(x) is: 
    A [–1, 1] Correct Answer Incorrect Answer
    B [–2, 2] Correct Answer Incorrect Answer
    C (–1, 1) Correct Answer Incorrect Answer
    D (–√2, √2) Correct Answer Incorrect Answer

    Solution

    We are given the function f(x) = cosx + sinx , and asked to find its range . To determine the range, observe that this is a standard form of a trigonometric identity. We can write: f(x) = cosx + sinx = √2{(cosx / √2) + (sinx/√2)} = √2 sin(x+π/4)
    This uses the identity: Where tan ϕ = b/a. In our case, a = 1, b = 1, so: f(x) = √2 sin(x+π/4)
    Since sin(θ) always lies in [−1,1], it follows that: f(x) = √2 sin(x+π/4) ∈ [−2​,2​].

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