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Problem Analysis

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Question 10

The question asks us to find the characteristics (properties) of the functions given in I, II, and III, and match them to the correct options from $A$ to $E$.

Given:

1. I. $y = 2x^4$: This is an even function as $x^4$ is even.
2. II. $y = x^3 + 1$: This is an odd function if we ignore the constant; however, it does not have symmetry around the origin due to $+1$. Thus, it may be identified as increasing.
3. III. $y = \frac{2}{x-1}$: This function has an undefined point at $x = 1$, implying the y-axis does not intercept.

From the table, we can match:

• $I - B$ (Even function)
• $II - D$ (Increasing function)
• $III - E$ (Does not cross the y-axis)

Correct answer: C) I-B, II-C, III-E

Question 11

The question asks about transformations on the function $y = f(x)$ based on shifts.

1. $O_x$ axis shift by -3 units implies vertical shift downwards.
2. $O_y$ axis shift by +3 units implies horizontal shift.
3. Shift by vector $(-3, 3)$ implies a combination.

Matching:

• $I - A$
• $II - B$
• $III - D$

Correct answer: B) I-A, II-B, III-D

Would you like further details or have any questions about the solutions?

Related Questions

1. How would the transformations change if the shifts were in the opposite direction?
2. Can you determine the symmetry of functions with both odd and even powers?
3. What are the implications of undefined points in rational functions?
4. How do horizontal and vertical shifts affect the graph of a function?
5. How does adding a constant term affect the symmetry of polynomial functions?

Tip: Remember that transformations applied outside the function affect the y-axis, while those inside affect the x-axis.