Types Of Functions

We need to determine when the argument of the logarithm is positive. For the function , the quadratic expression inside the logarithm is always positive, as it is a perfect square: Since , the argument of the logarithm is always positive. The range of is determined by the behavior of the logarithmic function. Since the argument can take any positive value, the logarithm can take any value from to , inclusive. Therefore, the range of is:

Step 1: Understanding the Concept:
Range of is . Then .
Step 2: Detailed Explanation:
Since , minimum of denominator gives maximum of and vice versa.
Denominator max . Denominator min . So range is .
Step 3: Final Answer:
Option (E) is correct.

Step 1: Understanding the Concept:
For square root, inside : .
Step 2: Detailed Explanation:
.
Thus domain is .
Step 3: Final Answer:
Option (C).

Concept: Use substitution symmetry.

Step 1: Write .


Step 2: Simplify expression.
Let , then:

Step 3: Add.

Concept: For square roots, expression inside must be non-negative.

Step 1: Apply conditions.


Step 2: Combine.

Concept: Use trigonometric identities:

Step 1: Substitute function.


Step 2: Use identity.


Step 3: Simplify.

Concept: Inverse image means solving:

Step 1: Set equation.


Step 2: Simplify.


Step 3: Factorize.


Step 4: Solve.

Concept:
• is defined only when argument
• Denominator

Step 1: Log condition


Step 2: Denominator condition
So

Step 3: Combine conditions

Final Conclusion:

Concept:
• Domain of is

Step 1: Apply domain condition


Step 2: Solve inequality
Add 1: Divide by 2: Final Conclusion:

Concept:
• To find inverse, write and solve for
• Domain restriction helps choose correct branch
• Range of becomes domain of

Step 1: Rewrite function


Step 2: Let and solve for


Step 3: Apply domain restriction
Since , we take negative root:

Step 4: Find range of
So, range is

Step 5: Write inverse function
Final Conclusion: