Test: Range of a Function - Grade 12 MCQ

As it can be seen from the above figure,
Domain of cos^-1x = [-1, 1] and Range of cos^-1x = [0, π]

f(x) = x² - 3x + 4
f(x) = x² - 2 (3/2) x + (3/2)² - (3/2)² + 4(Completing the square)
f(x) = (x - 3/2)² + 7/4
Since, (x - 3/2)² > 0
Thus, 

Range = [7/4, ∞]

Given function, 

Domain = (-∞, ∞) \{-3}
Let,
where y is in the range of h(x).
⇒ (x + 3)y = x - 2
⇒ xy - x + 3y + 2 = 0
⇒ x(y - 1) = -(3y + 2)

y can take any real value but y ≠ 1
So, the range of h(x) is (-∞, 1) ∪ (1, ∞)
∴ The correct answer is option (b).

Concept:
The range of a function is the complete set of all possible resulting values of the dependent variable.
Open interval can also be written 

• (a, b) or ]a, b[
• [a, b) or [a, b[

Calculation:
Let

Domain of f(x) is (−∞,∞)

So, y > 0        [As −1 ≤ sin3x ≤ 1]
From (1), 2 − sin3x = 1/y
⇒ sin3x = 2 − 1/y
⇒  sin3x = 

For x to be real 

 

Given:
f(x) = x²
Calculation:
f(x) = x²
⇒ Range of f = [0, ∞] = R⁺
⇒ Positive real numbers 
∴ The range of f will be positive real numbers

Concept:

Period of a function:

• If a function repeats over at a constant period we say that is a periodic function.
• It is represented like f(x) = f(x + T), T is the real number and this is the period of the function.
• The period of sin x and cos x is 2π

Calculation:
To Find: Period of 4cos³ x - 3cos x
As we know 4cos³ x - 3cos x = cos 3x
Period of cos x is 2π
Therefore, the Period of cos 3x is 2π/3

Concept:
Range: The range of a function is the set of all possible values it can produce, i.e., all values of y for which x is defined.
Note:
The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.
Calculation:
Let, y = f(x) = 
⇒y(x - 3) = x + 1
⇒yx - 3y - x = 1
⇒ x(y - 1) - 3y = 1
⇒ x(y - 1) = 1 + 3y

It is clear that x is not defined when y - 1 = 0, i.e, when  y = 1
∴ Range (f) = R - {1}
Hence, option (b) is correct.

Concept:
The domain of a relation R = {(x, y)} is the set of all values of x, and the range of R is the set of all values of y.
Calculation:
Since R is a relation on ℕ, the elements x and y must be positive integers.
We have x + 2y = 8.

For y to be positive and an integer (y ∈ N), we conclude that x must be divisible by 2 and 4 - x/2 > 0.

The only numbers less than 8 which are divisible by 2 are 2, 4 and 6.
∴ x ∈ {2, 4, 6} which is the required domain.

Concept:
The cartesian coordinate obtained by measuring parallel to the y-axis. 
Calculation:
Given: y = 2 - sin x  
The function sin x has all real numbers in its domain, but Range is -1 ≤ sin x ≤ 1 or  -1 ≤ -sin x ≤ 1. 
 -1 ≤ -sin x ≤ 1
Adding 2 both sides, we get
⇒ 2 - 1 ≤ 2 - sin x ≤ 2 + 1
⇒ 1 ≤ y ≤ 3
∴ y ∈ [ 1 , 3 ] 
The correct option is A.

Concept:
The number e, it is called the Euler's number, is an important mathematical constant approximately equal to 2.71828.
The approx. value of e is 2.71828
∴ 2 < e < 3