Which Of The Following Relations Are Functions?

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1. Which of the following relations are functions? I. y = -x + 2 II. x = -y + 2 III. x - y = 2

2. Find the range for f(x) = -x^2 + 1, for x > 0.

3. Find the domain for the given function options.

4. Find the range of the function: f(x) = x + 5, for x ≠ 2.

5. Determine whether f(x) = 5x^2 + 3x + 4 has a maximum or minimum.

6. Where is the function 4(x + 4)(x - 6)^3 > 0?

7. For which x-value would the graph of y = x^2 - 25 be below the x-axis?

8. Find f(g(-4)) if g(x) = (x + 5)^2.

9. Find f[g(x)] if f(x) = x^4 + 1 and g(x) = x^2.

10. Find g(x) if g(x) is the resulting function from moving f(x) = (x + 1) right 2 units and up 5 units.

11. Rewrite f(x) = sin(x) if the function is stretched vertically by a factor of 5.

12. What is the domain of the specified function?

13. Find the range of the function as given.

14. Is the function f(x) = |-4x| + x^4 even, odd, or neither?

15. Find the period and amplitude for f(x) = 2sin(3x).

16. Which one of the following is a function?

17. Determine the range of f(x) = (x - 2)^2 + 2.

18. Find the domain of f(x) = ...

19. Find the domain for the function f(x) = ...

20. Which of the following statements are true about functions and relations?

21. A box is to be constructed from a sheet of cardboard that is 20 cm by 50 cm by cutting out squares of length x from each corner and bending up the sides. What is the maximum volume of this box? Round your answer to two decimal places.

22. To three decimal places, find the value of the first positive x-intercept for f(x) = 2cos(x + 4).

23. Find the minimum value of f(x) = x^2 + 9x - 16.

24. Find (fg)(x) and (f + g)(x) given f(x) = 7x + 7 and g(x) = 6x^2.

25. Describe how the graph of y = x^2 can be transformed into the graph of the specified equation.

26. Describe how to transform the graph of f into the graph of g.

27. Describe the transformations required to obtain the graph of f(x) from g(x).

28. Determine the domain of the specified function.

29. Use the graph of f to estimate the local maximum and local minimum points.

30. Determine algebraically whether the function is even, odd, or neither.

31. State the vertical asymptote of the rational function f(x) = (x^2 + 3x + 2) / (x^2 + 5x + 4).

32. Provide examples and explanations related to polynomial roots and x-intercepts.

Paper For Above instruction

The set of problems presented encompasses a broad spectrum of fundamental concepts in algebra and functions. These problems aim to evaluate understanding of relations, functions, domain and range determination, transformations, and specific properties such as evenness or oddness, maxima, minima, and asymptotes. The following discussion will systematically analyze each aspect, providing comprehensive explanations grounded in mathematical principles.

Identifying Functions and Relations

Relations are any set of ordered pairs, but functions are relations where each input has exactly one output. For example, the relation y = -x + 2 is a function because for each x, y is uniquely determined. Similarly, the relation x = -y + 2 can be tested for function status via the vertical line test; it is a function if a vertical line intersects the graph at most once. The relation x - y = 2 can be rearranged into y = x - 2, which confirms the relation is a function, since each x-value corresponds to a unique y-value.

Finding Domain and Range

The domain of a function includes all possible x-values; in the case of f(x) = -x^2 + 1 for x > 0, the domain is (0, ∞). The range for this quadratic is y ≤ 1, but since x > 0, the maximum y-value 1 is achieved at x = 0 (not in the domain), hence the range is y

Maximum and Minimum Values

Quadratic functions like f(x) = 5x^2 + 3x + 4 are analyzed through the leading coefficient. Since it is positive, the parabola opens upward, indicating a minimum point. Calculating the vertex x = -b/2a = -3/(2*5) = -0.3, the minimum occurs at x ≈ -0.3. Substituting back gives the minimum value, confirming the parabola's minimum.

Inequalities and Solution Intervals

For the inequality 4(x + 4)(x - 6)^3 > 0, we analyze the sign of factors. Critical points are at x = -4 and x = 6. Testing intervals shows the inequality holds for x 6. This is a typical sign analysis to determine solution intervals of polynomial inequalities.

Graphical Considerations

Understanding the graph of y = x^2 - 25 reveals that the parabola opens upward with roots at x = ±5. The graph is below the x-axis when x is between -5 and 5. For the function y = 2sin(3x), the amplitude is 2, which is the coefficient in front of the sine function, and the period T = 2π / 3. These can be derived from the general sine function y = A sin(Bx), where amplitude = |A| and period = 2π / |B|.

Transformation of Functions

Transformations such as shifts, stretches, and reflections alter the basic graph. Horizontal shifts are achieved by adding or subtracting inside the function, e.g., y = (x - h)^2 shifts right by h. Vertical stretches by factors multiply the entire function, such as y = 5sin(x), which stretches the sine wave vertically by 5. Combinations of reflections and shifts further modify the graph, and understanding these helps in predicting the new locations and shapes of transformed functions.

Asymptotes and Polynomial Behavior

Rational functions like f(x) = (x^2 + 3x + 2) / (x^2 + 5x + 4) have vertical asymptotes where the denominator is zero, at x = -4 and x = -1, since division by zero is undefined. Zeroes are found where numerator equals zero, excluding points where the denominator is also zero, to avoid removable discontinuities. These asymptotes guide the end behavior and the shape of the graph.

Concluding Remarks

Mastery of these concepts enables students to analyze, manipulate, and graph functions with confidence. Applying algebraic, geometric, and analytical methods allows for a comprehensive understanding of the behavior of various types of functions and relations, fundamental to higher mathematics and its applications.

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