Domain Range Asymptotes Quiz: Domain, Range, And Vertical Asymptotes

1) The set of zeros of y = tanθ is {θ = kπ | k ∈ ℤ}.

True

False

tanθ = 0 when sinθ = 0 and cosθ ≠ 0. sinθ = 0 at θ = kπ, and cos(kπ) ≠ 0, so all such points are zeros.

Explanation

tanθ = 0 when sinθ = 0 and cosθ ≠ 0. sinθ = 0 at θ = kπ, and cos(kπ) ≠ 0, so all such points are zeros.

2) Select all true statements about y = tanθ.

Its period is π.

It has vertical asymptotes where cosθ = 0.

Its range excludes 0.

It is an odd function.

Its domain is all real θ.

Period of tan is π (tan(θ+π)=tanθ). It is undefined where cosθ=0, which are vertical asymptotes. tan is odd since tan(−θ)=−tanθ. The range includes 0; the domain excludes asymptotes.

Explanation

Period of tan is π (tan(θ+π)=tanθ). It is undefined where cosθ=0, which are vertical asymptotes. tan is odd since tan(−θ)=−tanθ. The range includes 0; the domain excludes asymptotes.

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3) Which statement best describes the behavior of tanθ near θ = π/2 from the left?

Tanθ → +∞

Tanθ → −∞

Tanθ → 0

Tanθ → 1

As θ approaches π/2 from the left (Quadrant I), sinθ → 1 and cosθ → 0+, so sinθ/cosθ → +∞.

Explanation

As θ approaches π/2 from the left (Quadrant I), sinθ → 1 and cosθ → 0+, so sinθ/cosθ → +∞.

4) Select all θ in [0, 2π) where tanθ is undefined.

π/2

π

3π/2

0

π/4

cosθ = 0 at θ = π/2 and 3π/2, so tanθ is undefined there. At 0, π, cosθ ≠ 0 and tanθ is defined.

Explanation

cosθ = 0 at θ = π/2 and 3π/2, so tanθ is undefined there. At 0, π, cosθ ≠ 0 and tanθ is defined.

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5) Which interval shows one full branch (one monotonic piece) of the basic tangent graph?

θ ∈ (−π/2, π/2)

θ ∈ (0, 2π)

θ ∈ (−π, π)

θ ∈ (π/2, 3π/2) including endpoints

Between two consecutive asymptotes, such as (−π/2, π/2), tanθ is continuous and monotone. Endpoints are excluded because the function is undefined there.

Explanation

Between two consecutive asymptotes, such as (−π/2, π/2), tanθ is continuous and monotone. Endpoints are excluded because the function is undefined there.

6) List all vertical asymptote locations for y = tanθ on [−2π, 2π].

Asymptotes occur at θ = π/2 + kπ. For k = −2, −1, 0, 1 we get −3π/2, −π/2, π/2, 3π/2 within [−2π, 2π].

Explanation

Asymptotes occur at θ = π/2 + kπ. For k = −2, −1, 0, 1 we get −3π/2, −π/2, π/2, 3π/2 within [−2π, 2π].

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7) Which statement best describes the behavior of tanθ near θ = π/2 from the right?

Tanθ → −∞

Tanθ → +∞

Tanθ → 0

Tanθ → 1

Approaching π/2 from the right (Quadrant II), sinθ → 1 and cosθ → 0−, so sinθ/cosθ → −∞.

Explanation

Approaching π/2 from the right (Quadrant II), sinθ → 1 and cosθ → 0−, so sinθ/cosθ → −∞.

8) Select all angles in (−π/2, π/2) where tanθ > 0.

−π/4

−π/6

11/13

π/6

π/4

In Quadrant I, tanθ>0. So π/6 and π/4 work. In Quadrant IV (negative angles), tanθ0.

Explanation

In Quadrant I, tanθ>0. So π/6 and π/4 work. In Quadrant IV (negative angles), tanθ0.

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9) Between θ = π/2 and θ = 3π/2, tanθ has exactly one zero.

True

False

The interval (π/2, 3π/2) spans one full period. tanθ is continuous and strictly increasing there and crosses zero once at θ = π.

Explanation

The interval (π/2, 3π/2) spans one full period. tanθ is continuous and strictly increasing there and crosses zero once at θ = π.

10) Select all that correctly describe the domain of y = tanθ.

All real θ except θ = π/2 + kπ

All real θ

All real θ except θ = kπ

All real θ except θ = π + 2kπ

All real θ except where cosθ = 0

The domain excludes the angles where tanθ is undefined, which are precisely where cosθ=0. Those angles are θ = π/2 + kπ.

Explanation

The domain excludes the angles where tanθ is undefined, which are precisely where cosθ=0. Those angles are θ = π/2 + kπ.

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11) On each interval between consecutive vertical asymptotes, y = tanθ is continuous and strictly increasing.

True

False

The derivative d/dθ(tanθ)=sec^2θ is positive wherever defined, so tanθ is strictly increasing and continuous on each interval between asymptotes.

Explanation

The derivative d/dθ(tanθ)=sec^2θ is positive wherever defined, so tanθ is strictly increasing and continuous on each interval between asymptotes.

12) Select all correct statements about asymptote locations.

They occur at θ = π/2 + kπ.

They are spaced π units apart.

They coincide with the zeros of cosθ.

They are the same as the zeros of tanθ.

None occur at θ = kπ.

cosθ has zeros at θ = π/2 + kπ. These are vertical asymptotes for tanθ, separated by π. tanθ zeros occur at kπ, not asymptotes, so D is false; E is true.

Explanation

cosθ has zeros at θ = π/2 + kπ. These are vertical asymptotes for tanθ, separated by π. tanθ zeros occur at kπ, not asymptotes, so D is false; E is true.

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13) Complete: tanθ is undefined when cosθ = ____.

Because tanθ = sinθ/cosθ, division by zero occurs exactly when cosθ = 0.

Explanation

Because tanθ = sinθ/cosθ, division by zero occurs exactly when cosθ = 0.

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14) Where does y = tanθ have vertical asymptotes?

θ = π/2 + kπ

θ = kπ

θ = π + 2kπ

θ = 0

tanθ = sinθ/cosθ is undefined when cosθ = 0. cosθ = 0 at θ = π/2 + kπ for any integer k, which are the vertical asymptote locations.

Explanation

tanθ = sinθ/cosθ is undefined when cosθ = 0. cosθ = 0 at θ = π/2 + kπ for any integer k, which are the vertical asymptote locations.

15) State the domain of y = tanθ in set-builder form.

tanθ is undefined when cosθ = 0, i.e., at θ = π/2 + kπ. Excluding these gives the domain: ℝ \ {π/2 + kπ}.

Explanation

tanθ is undefined when cosθ = 0, i.e., at θ = π/2 + kπ. Excluding these gives the domain: ℝ \ {π/2 + kπ}.

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16) The range of y = tanθ is all real numbers.

True

False

Between any two vertical asymptotes, tanθ increases continuously from −∞ to +∞, so every real value is attained in each period; hence the range is (−∞, ∞).

Explanation

Between any two vertical asymptotes, tanθ increases continuously from −∞ to +∞, so every real value is attained in each period; hence the range is (−∞, ∞).

17) Which is the correct general formula for the domain of y = tanθ?

θ ∈ ℝ \ {π/2 + kπ}

θ ∈ ℝ \ {kπ}

θ ∈ ℝ \ {π + 2kπ}

θ ∈ ℝ only

Exclude the infinite set of asymptote points where cosθ=0: θ = π/2 + kπ. All other real θ are allowed.

Explanation

Exclude the infinite set of asymptote points where cosθ=0: θ = π/2 + kπ. All other real θ are allowed.

18) As θ approaches any vertical asymptote of y = tanθ from the left, tanθ always tends to +∞.

True

False

The sign depends on the quadrant. For example, as θ→π/2−, tanθ→+∞, but as θ→3π/2− (Quadrant III), tanθ→−∞. So it is not always +∞.

Explanation

The sign depends on the quadrant. For example, as θ→π/2−, tanθ→+∞, but as θ→3π/2− (Quadrant III), tanθ→−∞. So it is not always +∞.

19) State the range of y = tanθ.

On each open interval between asymptotes, tanθ takes every real value from −∞ to +∞ exactly once.

Explanation

On each open interval between asymptotes, tanθ takes every real value from −∞ to +∞ exactly once.

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20) Which description matches the graph of y = tanθ near θ = 3π/2 from the left?

It increases without bound (→ +∞).

It decreases without bound (→ −∞).

It approaches 0.

It oscillates between −1 and 1.

As θ→3π/2−, sinθ→−1 and cosθ→0−. Then tanθ = (−1)/(a small negative) → (+∞).

Explanation

As θ→3π/2−, sinθ→−1 and cosθ→0−. Then tanθ = (−1)/(a small negative) → (+∞).

Domain Range Asymptotes Quiz: Domain, Range, and Vertical Asymptotes