If R is the set of real numbers and f : R → R is a function defined by f ( x ) = sin x , then f - 1 ( [ - 1 , 1 ] ) is

{ x ∣ x = n π , n is an integer }

{ x ∣ x = π / 2 + 2 n π , n is an integer}

Let f x = ( x + 1 ) 2 - 1 , x ≥ - 1 . Then x ∣ f x = f - 1 x =

- 1 , - 3 + i 3 2 , - 3 - i 3 2

Which of the following functions has inverse function

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

R

is the set of real numbers and

f : R \to R

is a function defined by

f (x) = \sin x,

then

f^{- 1} ([- 1, 1])

is

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Directions: Each of the following sentences in this section has a blank space and four words or group of words given after the sentence. Select the word or group of words you consider most appropriate for the blank space and indicate your response on the Answer Sheet accordingly.

Every rash driver becomes a ____ killer.

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let $f : X \to Y$ and $A, B$ are non-void subsets of $Y,$ then (where the symbols have their usual interpretation)

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

The inverse of

y = 5^{\log x}

is:

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

The inverse of the function

f (x) = \log_{a} (x + \sqrt{x^{2} + 1})

(where,

a < 0, a \neq 1

) is

HARD

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

f : [0, \infty) \to [2, \infty)

is given by

f (x) = x + \frac{1}{x}

, then

f^{- 1} (x)

equals

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

f : R \to R, g : R \to R

are two functions defined by

f (x) = 2 x - 3, g (x) = x^{3} + 5

, then

{(f o g)}^{- 1} (x) =

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

f (x) = (x + 1)^{2} - 1, x \geq - 1

, then the set

\{x : f (x) = f^{- 1} (x)\}

is equal to

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Two sentences are given below and you are required to find the correct sentence which combines both the sentences.

Which is the correct combination of the given two sentences?

He is too tired. He could not stand.

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

f : [1, \infty) \to [1, \infty]

is defined by

f (x) = \frac{1 + \sqrt{1 + 4 \log_{2} x}}{2}

, then

f^{- 1} (3) =

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

In the following sentence, a blank space with four options is given. Select whichever preposition or determiner you consider the most appropriate for the blank space.

These are the good rules to live _____.

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

The inverse function of

f (x) = \frac{8^{2 x} - 8^{- 2 x}}{8^{2 x} + 8^{- 2 x}}, x \in (- 1, 1),

is __________.

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let

f : R - {3} \to R - {1}

be defined by

f (x) = \frac{x - 2}{x - 3}

. Let

g : R \to R

be given as

g (x) = 2 x - 3

. Then, the sum of all the values of

x

for which

f^{- 1} (x) + g^{- 1} (x) = \frac{13}{2}

is equal to

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let

f : N \to Y

(

N

is the set of natural numbers) defined as

f (x) = 4 x + 3

where

Y = {y \in N; y = 4 x + 3

for some

x \in N}

. Then the inverse of

f (x)

is equal to

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

f : R - \{\frac{3}{5}\} \to R - \{\frac{3}{5}\}; f (x) = \frac{3 x + 1}{5 x - 3},

then ________.

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let

f (x) = x^{2} - x + 1, x \geq \frac{1}{2},

then the solution of the equation

f (x) = f^{- 1} (x)

is

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

If

f (x) = \frac{3 x + 2}{5 x - 3},

then

HARD

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let

f (x) = (x + 1)^{2} - 1, x \geq - 1 .

Then

\{x ∣ f (x) = f^{- 1} (x)\} =

MEDIUM

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Let

f : R \to R, g : R \to R

be two functions such that

f (x) = 2 x - 3, g (x) = x^{3} + 5

. The function

{(f o g)}^{- 1} (x)

is equal to

EASY

Mathematics>Functions>Building Functions>Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain

Consider the function

y = \log_{a} (x + \sqrt{x^{2} + 1})

a > 0, a \neq 1

. The inverse of the function

Which of the following functions has inverse function?

Important Questions on Building Functions

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