Graphical Analysis of Range of Sine Functions
The range of a function \( y = f(x) \) is the set of values \( y \) takes for all values of \( x \) within the domain of \( f \).
What is the range of \( y = f(x) = \sin(x) \)?
The domain of \( f \) above is the set of all values of \( x \) in the interval \( (-\infty, +\infty) \).
As \( x \) takes values from \( -\infty \) to \( +\infty \), \( \sin(x) \) takes all values between -1 and 1. Hence:
\( -1 \leq \sin(x) \leq 1 \quad \text{or} \quad -1 \leq y \leq 1 \)
In general, the range of any sine function of the form \( y = \sin(bx + c) \) is given by:
\( -1 \leq \sin(bx + c) \leq 1 \quad \text{or} \quad -1 \leq y \leq 1 \)
Examples with Solutions
Example 1
Find the range of function \( f \) defined by:
\[ f(x) = -\sin(x) \]View Solution
Start with the range of the basic sine function:
\[ -1 \leq \sin(x) \leq 1 \]Multiply all terms of the inequality by -1 and reverse the inequality symbols to obtain:
\[ 1 \geq -\sin(x) \geq -1 \quad \text{or} \quad -1 \leq -\sin(x) \leq 1 \]Hence, the range of \( -\sin(x) \) is given by the interval:
\[ [-1, 1] \]Example 2
Find the range of function \( f \) defined by:
\[ f(x) = 2\sin\left(-3x - \frac{\pi}{6}\right) \]View Solution
The range of \( \sin\left(-3x - \frac{\pi}{6}\right) \) is given by:
\[ -1 \leq \sin\left(-3x - \frac{\pi}{6}\right) \leq 1 \]Multiply all terms of the inequality by 2 to obtain:
\[ -2 \leq 2\sin\left(-3x - \frac{\pi}{6}\right) \leq 2 \]The range of the given function \( f \) in interval form is:
\[ [-2, 2] \]Example 3
Find the range of function \( f \) defined by:
\[ f(x) = 0.1\sin\left(\frac{x}{\pi} + \pi\right) - 2 \]View Solution
The range of \( \sin\left(\frac{x}{\pi} + \pi\right) \) is given by:
\[ -1 \leq \sin\left(\frac{x}{\pi} + \pi\right) \leq 1 \]Multiply all terms of the inequality by 0.1 to obtain:
\[ -0.1 \leq 0.1\sin\left(\frac{x}{\pi} + \pi\right) \leq 0.1 \]Add -2 to all terms of the above inequality to obtain:
\[ -2.1 \leq 0.1\sin\left(\frac{x}{\pi} + \pi\right) - 2 \leq 1.9 \]The range of values may be written in interval form as follows:
\[ [-2.1, 1.9] \]