Find the Domain and Range f(x)=sec(x)

Problem

ƒ(x)=sec(x)

Solution

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

sec(x)=1/cos(x)

2. Determine the domain by finding where the denominator is non-zero.

cos(x)≠0

3. Solve for the values where the cosine function equals zero.

x≠π/2+n*π

where *n∈ℤ

4. Analyze the range of the cosine function to determine the range of its reciprocal.

−1≤cos(x)≤1

5. Apply the reciprocal property, noting that as cos(x) approaches zero, sec(x) approaches positive or negative infinity, and the values between −1 and 1 (excluding zero) result in magnitudes greater than or equal to 1

sec(x)≥1* or *sec(x)≤−1

Final Answer

Domain: *{x|x≠π/2+n*π,n∈ℤ}, Range: *(−∞,−1]∪[1,∞)

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