Question

Find the range of the function f(x) = `1/(3−2 sinx)`

Answer 1

Given:

f(x) = `1/(3 − 2 sin x)`

Step 1: Range of sin⁡x

−1 ≤ sinx ≤ 1

Step 2: Find the range of denominator

3 − 2 sin x

Multiply inequality by −2 (reverse signs):

−2 ≤ −2 sin x ≤ 2

1 ≤3 − 2 sinx ≤ 5    ...[So, denominator lies in: (1, 5)]

Step 3: Find the range of f(x)

Since f(x) = `1/"denominator"`, and the denominator is positive

Minimum value of f(x) occurs when the denominator is maximum (5)

The maximum value of f(x) occurs when the denominator is minimum (1)

`1/5 ≤ f(x) ≤ 1`

`[1/5, 1]`