Question

Find the domain and range of √(9x-x²) ​

Answer 1

EXPLANATION.

Domain and range of fυnction.

⇒ √(9x - x²).

As we know that,

⇒ 9x - x² ≥ 0.

⇒ x(9 - x) ≥ 0.

⇒ x(x - 9) ≤ 0.

First we find the zeroes, we get.

⇒ x = 0 and x = 9.

Put this point on wavy curve method, we get.

⇒ x ∈ [0,9].

Domain of the fυnction,

⇒ √9x - x² = x ∈ [0,9].

For  of the equation,

Let the fυnction = y.

Equation is written as,

⇒ y = √9x - x².

⇒ y² = 9x - x².

⇒ y² - 9x + x² = 0.

⇒ x² - 9x + y² = 0.

Make a quadratic equation on x.

Quadratic equation must be : D ≥ 0.

⇒ b² - 4ac ≥ 0.

⇒ (-9)² - 4(1)(y²) ≥ 0.

⇒ 81 - 4y² ≥ 0.

⇒ 4y² - 81 ≤ 0.

As we know that,

Formula of :

⇒ (x² - y²) = (x + y)(x - y).

⇒ (2y)² - (9)² = (2y + 9)(2y - 9).

⇒ (2y + 9)(2y - 9) ≤ 0.

Find zeroes of the equation, we get.

⇒ y = -9/2 and y = 9/2.

Put this zeroes on wavy curve method, we get.

⇒ y ∈ [-9/2, 9/2]. = (1).

As we also know that,

Equation = √9x - x² is defined on interval of [0,∞). = (2).

Taking intersection on both equation (1) & (2), we get.

⇒ y ∈ [-9/2, 9/2] ∩ [0,∞).

⇒ y ∈ [0,9/2].

Range of the equation,

⇒ √9x - x² = y ∈ [0,9/2].

Answer 2

Answer:

Domain:

» f = √(9x - x²)

The function will be defined for 9x - x² ≥ 0

i.e. x (9 - x) ≥ 0

i.e. x ≤ 9 and x ≥ 0

Therefore, the domain is [0, 9]

Range:

y = √(9x - x²)

y = 0, is the least value possible, and max value will be possible for max value of 9x - x²

i.e. max value of x (9 - x) where x € [0, 9]

» 4.5 (9 - 4.5) = 20.25

Therefore, max range of y = √(20.25) = 4.5 = 9/2

Hence, range is [0, 9/2]