Find the Range y=sec(x)

Problem

y=sec(x)

Solution

1. Identify the definition of the secant function in terms of the cosine function.

sec(x)=1/cos(x)

2. Determine the range of the cosine function, which is the denominator of the expression.

−1≤cos(x)≤1

3. Analyze the behavior of the reciprocal function. Since cos(x) takes values between −1 and 1 (excluding 0 where the secant is undefined), its reciprocal will result in values that are either greater than or equal to 1 or less than or equal to −1

4. Consider the positive interval where 0<cos(x)≤1 Taking the reciprocal gives:

sec(x)≥1

5. Consider the negative interval where −1≤cos(x)<0 Taking the reciprocal gives:

sec(x)≤−1

6. Combine these two intervals to express the full range of the function.

(−∞,−1]∪[1,∞)

Final Answer

Range of *sec(x)={y∈ℝ:y≤−1* or *y≥1}

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