TRANSCENDENTAL FUNCTIONS

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Test your understanding of transcendental functions, including exponential, logarithmic, and trigonometric functions. This quiz covers key concepts, graphs, properties, and applications to help reinforce your knowledge and problem-solving skills.

1 / 30

The function

$f(x) = e^x$

has a range of:

(-infty, infty)

(0, infty)

[0, infty)

(-infty, 0)

$e^x$

is always positive for all real

$x$

2 / 30

The inverse of

$f(x) = 2^x$

is:

ln x

log_x 2

log_2 x

e^x

Solve

$y = 2^x$

for

$x$

to get

$x = log_2 y$

3 / 30

The derivative of

$f(x) = 3e^{2x}$

is:

6e^{2x}

3e^{2x}

2e^{2x}

e^{2x}

Use the chain rule:

$frac{d}{dx} e^{u} = e^{u} cdot u’$

4 / 30

The solution to

$e^{2x} = 5$

is:

x = ln 5

x = frac{5}{2}

x = e^{5/2}

x = frac{ln 5}{2}

Take the natural log of both sides and solve for

$x$

5 / 30

The function

$f(x) = e^{-x}$

decays as

$x$

increases because:

The base

e

is positive

It is an even function

The exponent is negative

It is periodic

$e^{-x} = frac{1}{e^x}$

, which decreases as

$x$

increases.

6 / 30

The horizontal asymptote of

$f(x) = 1 – e^x$

is:

y = 0

y = e

y = 1

y = -1

$x to -infty$

$e^x to 0$

, so

$f(x) to 1$

7 / 30

The value of

$e^{ln 3}$

is:

3

e^3

ln 3

Undefined

$e^{ln x} = x$

for

$x > 0$

8 / 30

The function

$f(x) = 2^{x+1}$

can be rewritten as:

2^x + 1

(x+1)^2

e^{(x+1)ln 2}

2 cdot 2^x

$2^{x+1} = 2^x cdot 2^1$

9 / 30

The growth rate of

$f(x) = e^{kx}$

is fastest when:

k

is small and positive

k

is large and positive

k

is negative

k = 0

Larger

$k$

values lead to steeper exponential growth.

10 / 30

The solution to

$4e^{3x} = 20$

is:

x = frac{ln 20}{4}

x = ln 5

x = frac{ln 5}{3}

x = frac{3}{ln 5}

11 / 30

The domain of

$f(x) = ln(x – 2)$

is:

x > 2

x geq 2

x < 2

All real numbers

12 / 30

The value of

$log_2 16$

is:

2

8

16

4

$2^4 = 16$

13 / 30

The equation

$ln x = 3$

has the solution:

x = 3^e

x = e^3

x = 10^3

x = ln 3

Rewrite in exponential form:

$x = e^3$

14 / 30

The expression

$log_3 27 – log_3 9$

simplifies to:

2

3

1

0

$log_3 left( frac{27}{9} right) = log_3 3 = 1$

15 / 30

The derivative of

$f(x) = ln(5x)$

is:

frac{1}{x}

frac{1}{5x}

frac{5}{x}

ln 5

$frac{d}{dx} ln(ax) = frac{1}{x}$

for

$a > 0$

16 / 30

The function

$f(x) = log_4 x$

is equivalent to:

frac{ln 4}{ln x}

ln(x^4)

e^{4x}

frac{ln x}{ln 4}

Change of base formula:

$log_b a = frac{ln a}{ln b}$

17 / 30

The vertical asymptote of

$f(x) = ln(x + 1)$

is:

x = 0

x = -1

x = 1

y = 0

The argument

$x + 1$

must be positive, so

$x to -1^+$

18 / 30

The solution to

$log_5 (x + 3) = 2$

is:

x = 22

x = 10

x = 7

x = 25

Rewrite as

$x + 3 = 5^2$

, so

$x = 25 – 3 = 22$

19 / 30

The expression

$ln(e^2 cdot e^3)$

simplifies to:

6

e^5

5

2 ln 3

$ln(e^2 cdot e^3) = ln(e^{2+3}) = 5$

20 / 30

The function

$f(x) = ln x$

and

$g(x) = e^x$

are inverses because:

Both are increasing

f(g(x)) = x

and

g(f(x)) = x

Both are transcendental

They have the same domain

Inverse functions satisfy

$f(g(x)) = x$

and

$g(f(x)) = x$

21 / 30

The period of

$f(x) = sin(2x)$

is:

2pi

frac{pi}{2}

4pi

pi

The period of

$sin(Bx)$

$frac{2pi}{B}$

22 / 30

The amplitude of

$f(x) = -3 cos x$

is:

3

-3

1

pi

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The exact value of

$sinleft(frac{pi}{6}right)$

is:

frac{sqrt{3}}{2}

frac{1}{2}

frac{sqrt{2}}{2}

1

24 / 30

The phase shift of

$f(x) = 2 sin(3x – pi)$

is:

pi

to the left

No phase shift

frac{pi}{3}

to the right

frac{pi}{3}

to the left

Rewrite as

$2 sinleft(3left(x – frac{pi}{3}right)right)$

25 / 30

The range of

$f(x) = tan x$

is:

(-infty, infty)

[-1, 1]

[0, infty)

(-infty, 0]

26 / 30

The identity

$sin^2 x + cos^2 x$

equals:

0

sin(2x)

cos(2x)

1

27 / 30

The maximum value of

$f(x) = 4 + 2 sin x$

is:

4

6

2

0

Maximum of

$sin x$

is 1, so

$4 + 2(1) = 6$

28 / 30

The general solution to

$sin x = frac{1}{2}$

is:

x = frac{pi}{3} + 2pi n

x = frac{pi}{2} + pi n

x = frac{pi}{6} + 2pi n

x = frac{5pi}{6} + 2pi n

x = frac{pi}{4} + pi n

29 / 30

The derivative of

$f(x) = cos(3x)$

is:

-3 sin(3x)

3 sin(3x)

-sin(3x)

sin(3x)

Use the chain rule:

$frac{d}{dx} cos(u) = -sin(u) cdot u’$

30 / 30

The function

$f(x) = sec x$

is undefined when:

sin x = 0

tan x = 0

cot x = 0

cos x = 0

$sec x = frac{1}{cos x}$

, which is undefined where

$cos x = 0$

Your score is

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