Question

If function \(f(x)=\log _{2}\left(\frac{\sin x-\cos x+3 \sqrt{2}}{\sqrt{2}}\right)\), then (1) Domain of \(f(x)\) is \(R\) (2) Domain of \(f(x)\) is \(R-\\{n \pi\\}, n \in I\) (3) Range of \(f(x)\) is \([0,1]\) (4) Range of \(f(x)\) is \(R-[1,2]\)

Short answer

Domains: \( R - \{n \pi\} \), Range: \( R \)

Step-by-step solution

Title - Understand the Logarithmic Function

For the logarithmic function \(f(x) = \log_{2}\left(\frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}}\right)\), the argument inside the logarithm (\( \frac{\sin x - \cos x + 3 \sqrt{2}}{\sqrt{2}} \)) must be greater than zero.

Title - Simplify the Argument

Express the argument inside the logarithm in a simpler form. We need \(\frac{\sin x - \cos x + 3 \sqrt{2}}{\sqrt{2}} > 0\). Start by isolating \(\sin x - \cos x\): \[ \sin x - \cos x + 3 \sqrt{2} > 0. \]

Title - Factor Trigonometric Terms

Recall the range of the function \(\sin x - \cos x\), which can be expressed as: \[ (\sin x - \cos x) = \sqrt{2} \sin (x - \frac{\pi}{4}) \] This expression ranges from \( \- \sqrt{2} \) to \( \sqrt{2} \).

Title - Determine Domain of Argument

For the inequality \[ \sqrt{2} \sin (x - \frac{\pi}{4}) + 3 \sqrt{2} > 0, \ \sin (x - \frac{\pi}{4}) > -3. \] Since the sine function ranges from -1 to 1, the inequality is always satisfied. Therefore, no restrictions are required, so all real numbers (not multiples of \( \pi \)) satisfy the inequality.

Title - Analyze the Range

The function \( f(x) \) is a logarithmic function. If the input to the logarithm varies from 0 to 4\(\sqrt{2} \), then the range for \( \log_{2}(x) \) must be determined. However, remembering the properties of logarithmic functions, \( f(x) \) spans \ all real numbers. \( \)

Key concepts

Domain of Functions

The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function \( f(x) = \log_{2}\left(\frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}}\right) \), we need to ensure that the argument inside the logarithm is greater than zero.

The argument in our function is \( \frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}} \). So, we need: \[ \frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}} > 0 \]

To simplify, let's isolate \(\sin x - \cos x\): \[ \sin x - \cos x + 3\sqrt{2} > 0 \] By knowing that \sin x - \cos x ranges from -\sqrt{2} to \sqrt{2}, adding 3\sqrt{2} means it will always be positive. Thus, this inequality holds for all real numbers, except at certain points where the expression is undefined, where \sin x = \cos x.

Therefore, the domain of \( f(x) \) will be all real numbers except for multiples of \pi.

Range of Functions

The range of a function is the set of all possible output values (y-values) the function can produce. For our given function \( f(x) = \log_{2}\left(\frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}}\right) \), we need to calculate the possible output values from the input we determined.

Since the function inside the logarithm is positively valued and can span from 0 to a large number, the transformation applied by the logarithm on this input will essentially mean our range spans all real numbers.

Another way to think about it is remembering the properties of the logarithmic functions—since as input increases positively from 0, the logarithmic values transition from large negative numbers to large positive numbers.

Trigonometric Inequalities

Understanding trigonometric inequalities is crucial to solving problems involving them. For the inequality within our given function, \(\frac{\sin x - \cos x + 3\sqrt{2}}{\sqrt{2}} > 0\), it involves the sine and cosine functions which are periodic and range between -1 and 1.

Recall that \(\sin x - \cos x\) can be written as \(\sqrt{2} \sin (x - \frac{\pi}{4}) \). This transformation shows that it ranges between -\sqrt{2} and \sqrt{2}.

By adding 3\sqrt{2}, we'll always have a value greater than zero, provided that there are no points where the sine function makes the denominator zero.
Hence, addressing trigonometric inequalities often involves reducing or transforming the expressions into simpler forms to glean their domains and ranges, ensuring positive values for logarithmic applications.