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Understanding Key Concepts in Mathematics: Functions, Domains, and Graphs

Explore key mathematical concepts including linear functions, quadratic equations, and rational expressions. Analyze domains and ranges, vertical and horizontal asymptotes, and the relationship between lines, such as perpendicularity. Learn to solve equations, factor expressions, and find key points like vertices and intercepts. This resource provides step-by-step explanations, sample problems, and helps build a solid foundation in algebraic principles essential for advanced studies.

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Understanding Key Concepts in Mathematics: Functions, Domains, and Graphs

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  1. Linear & Square Roots Quadratics Rationals Exponentials & Logs Inverses $100 $100 $100 $100 $100 $200 $200 $200 $200 $200 $300 $300 $300 $300 $300 $400 $400 $400 $400 $400 $500 $500 $500 $500 $500

  2. NC State the domain & range:

  3. Domain: x > 8 Range: y < 9

  4. What is the relationship between these lines?

  5. The lines are perpendicular.

  6. NC What is m(8) given the following?

  7. What is the rate of change for the following?

  8. Solve:

  9. X= 3, 5

  10. Factor:

  11. (6x + 1)(2x – 7)

  12. Find the vertex, y-intercept, and x-intercepts of the following:

  13. Vertex: (1.25, 6.125) Y-intercept: (0, 3) X-intercepts: (-0.5, 0) and (3, 0)

  14. What is the value of x when x > 0 and:

  15. X = 4

  16. What is the factored form for the following?

  17. (3x + 4)(-6x + 5)

  18. The area of a rectangular classroom is 35 square feet. The length of the room is 2 feet more than the width. Find the width of the rectangular classroom. Write an equation, show work, and explain your work.

  19. Equation: 35 = x(x + 2) Width = 5 feet

  20. Write the equation of the rational function in transformation form that has a vertical asymptote of x = -4 and a horizontal asymptote of y = 3.

  21. Solve:

  22. Find the vertical asymptote, horizontal asymptote, hole, zeros, y-intercept, and equation of the graph.

  23. Solve:

  24. Find the vertical asymptote, horizontal asymptote, hole, zeros, y-intercept, and equation of the graph.

  25. Simplify the following:

  26. Solve without a calculator:

  27. Without a calculator, solve for x. Your answer will be an expression.

  28. State the domain and range:

  29. Domain: x > 1 Range: all real numbers

  30. Solve:

  31. Solve for m.

  32. Given f(1) = 4, f(0) = -2, and that f(x) is a linear function, find the inverse equation.

  33. Find g-1(x).

  34. Find an equation for the inverse.

  35. Find the inverse.

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