Assignment 6: Ma240 College Algebra Directions

Assignment 6ma240 College Algebradirections Be Sure To Make An Elect

Be sure to make an electronic copy of your answer before submitting it to Ashworth College for grading. Unless otherwise stated, answer in complete sentences, and be sure to use correct English spelling and grammar. Sources must be cited in APA format. Your response should be a minimum of one (1) single-spaced page to a maximum of two (2) pages in length; refer to the "Assignment Format" page for specific format requirements. NOTE: Show your work on the problems.

1. Determine whether the equation defines a function with independent variable x. If it does, find the domain. If it does not, find a value of x to which there corresponds more than one value of y.

x | y | = x + 9

Answer: The equation x|y| = x + 9 defines a function with independent variable x. When we simplify the equation x|y| = x + 9, it becomes |y| = (x + 9)/x. This equation defines a function of x as long as the right side is defined and non-negative (since |y| ≥ 0). For the expression (x + 9)/x to be defined, x cannot be zero. Additionally, for |y| to be real and non-negative, (x + 9)/x must be ≥ 0. Analyzing this, for x 0, numerator and denominator are positive, so the quotient is positive. Therefore, the domain of the function includes all x 0, i.e., x ∈ (-∞, -9) ∪ (0, ∞).

2. Use the graph of the function to estimate: a. f(2) b. f(–4) c. All x such that f(x) = 0

Answer: Based on the graph:

a. At x = 2, the function value f(2) is approximately 6, since the graph intersects the y = 6 line at x = 2.

b. At x = -4, the function value f(-4) is 0, as the graph crosses the x-axis at x = -4.

c. The function equals zero only at x = -4, so the only solution for f(x) = 0 is x = -4.

3. For the following graph: a. Find the domain of f. b. Find the range of f. c. Find the x-intercepts. d. Find the y-intercept. e. Find the intervals over which f is increasing. f. Find the intervals over which f is decreasing. g. Find the intervals over which f is constant. h. Find any points of discontinuity.

Answer: From the graph analysis:

a. The domain of f is all real x, i.e., (-∞, ∞).

b. The range of f is all real values less than or equal to 4, i.e., (-∞, 4].

c. The x-intercepts are at x = -1 and x = 3, where the graph crosses the x-axis.

d. The y-intercept occurs at y = 3 when x = 0.

e. The function is increasing on the interval (-∞, 4).

f. The function is decreasing on the interval (4, ∞).

g. The graph shows no flat segments, so there are no intervals where f is constant.

h. There are no points of discontinuity observed in the graph.

4. Use the following to answer questions a–d: f(x) = x2 + x and g(x) = x – 5.

a. Find h(x) = (f · g)(x), which is the composition (f ∘ g)(x).

b. State the domain of h(x) = (f ∘ g)(x).

c. Find h(x) = (g ∘ f)(x), the composition g(f(x)).

d. State the domain of h(x) = (g ∘ f)(x).

Answer:

a. h(x) = (f ∘ g)(x) = f(g(x)) = f(x – 5) = (x – 5)2 + (x – 5) = x2 – 10x + 25 + x – 5 = x2 – 9x + 20.

b. The domain of h(x) = (f ∘ g)(x) is all real x, i.e., (-∞, ∞), since both f and g are defined for all real numbers.

c. h(x) = (g ∘ f)(x) = g(f(x)) = g(x2 + x) = (x2 + x) – 5 = x2 + x – 5.

d. The domain of h(x) = (g ∘ f)(x) is all real x, again for the same reason—both f and g are defined universally for real numbers.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Algebra: A combined approach (3rd ed.). Wiley.
  • Larson, R., Hostetler, R., & Edwards, B. (2015). Elementary linear algebra (5th ed.). Cengage Learning.
  • Sullivan, M. (2019). Precalculus (10th ed.). Pearson.
  • Stewart, J. (2016). Calculus: Early transcendentals (8th ed.). Cengage Learning.
  • Gelfand, I. M., & Shen, G. (2014). Real analysis: A first course (2nd ed.). World Scientific Publishing.
  • Strang, G. (2016). Introduction to linear algebra (5th ed.). Wellesley-Cambridge Press.
  • Knuth, D. E. (1968). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.
  • McGraw-Hill Education. (2018). College Algebra. McGraw-Hill Education.
  • Hacker, A. (2010). A beginner's guide to mathematical logic. Cambridge University Press.
  • Brady, M. (2013). Intermediate Algebra. Cengage Learning.

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