Find the Domain and Range f(x)=sec(2x)

Problem

ƒ(x)=sec(2*x)

Solution

1. Identify the definition of the secant function. The function sec(2*x) is defined as 1/cos(2*x)

2. Determine the domain by finding where the denominator is non-zero. The function is undefined when cos(2*x)=0

3. Solve for the excluded values of x The cosine function is zero at odd multiples of π/2

2*x=π/2+n*π

x=π/4+(n*π)/2

where n is any integer.

4. State the domain. The domain consists of all real numbers except the values found in the previous step.

Domain={[x∈ℝ,x≠π/4+(n*π)/2,n∈ℤ]}

5. Determine the range based on the properties of the secant function. The range of sec(θ) is (−∞,−1]∪[1,∞) Since the coefficient of the argument 2*x only affects the period and not the amplitude, the range remains the same.

Range=(−∞,−1]∪[1,∞)

Final Answer

Domain: *{[x∈ℝ,x≠π/4+(n*π)/2,n∈ℤ]}, Range: *(−∞,−1]∪[1,∞)

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