Math11

• What is the sum of two functions f and g represented as?


  (f + g)(x) = f(x) + g(x)
• How is the difference of two functions f and g represented?


  (f - g)(x) = f(x) - g(x)
• What is the product of two functions f and g represented as?


  (f ⋅ g)(x) = f(x) · g(x)
• How is the quotient of two functions f and g represented?


  (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
• What does it mean when g(x) = 0 in the context of the quotient of functions?

  It means the quotient is undefined.
• Given f(x) = x² - 1 and g(x) = x² - x, what is (f + g)(x)?


  (f + g)(x) = 2x² - x - 1
• What is the result of (f - g)(x) when f(x) = x² - 1 and g(x) = x² - x?

  (f - g)(x) = x - 1
• How do you compute (f ⋅ g)(x) for f(x) = x² - 1 and g(x) = x² - x?


  (f ⋅ g)(x) = x²(x² - x) - (x² - 1)(x² - x)
• What is the quotient (f / g)(x) for f(x) = x² - 1 and g(x) = x² - x?

  (f / g)(x) = (x + 1)/(x - 1)
• Given f(x) = x + 5, g(x) = 2x - 1, and h(x) = 2x² + 9x - 5, what is (f + g)(x)?

  (f + g)(x) = 3x + 4
• What is the result of (f - g)(x) when f(x) = x + 5 and g(x) = 2x - 1?

  (f - g)(x) = -x + 6
• How do you compute (f ⋅ g)(x) for f(x) = x + 5 and g(x) = 2x - 1?


  (f ⋅ g)(x) = 2x² + 9x - 5
• What is the composition of functions h(g(x)) when h(x) = 2x² + 9x - 5 and g(x) = 2x - 1?

  h(g(x)) = 2(2x - 1)² + 9(2x - 1) - 5
• What is the definition of a composite function?

  A composite function is a function formed by combining two functions, denoted as \( f(g(x)) \).
• What is the example of a function given in the study material?


  One example is \( f(x) = 3x - 4 \).
• How is the composite function \( f(g(x)) \) calculated in the example provided?

  It is calculated by substituting \( g(x) \) into \( f(x) \), resulting in \( f[g(x)] = 3(g(x)) - 4 \).
• What is the result of \( f[g(x)] \) when \( g(x) = x^2 - 3 \)?

  The result is \( 3(x^2 - 3) - 4 = 3x^2 - 9 - 4 = 3x^2 - 13 \).
• What is the second example of a function given in the study material?

  The second example is \( f(x) = 5x + 2 \).
• What are the operations that can be performed on functions?

  • Sum: (f + g)(x) = f(x) + g(x)
  • Difference: (f - g)(x) = f(x) - g(x)
  • Product: (f ⋅ g)(x) = f(x) · g(x)
  • Quotient: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
• What is the value of \( f(4) \) when \( f(x) = 5x + 2 \)?

  The value is \( 5(4) + 2 = 20 + 2 = 22 \).
• What does \( f/g(2) \) represent in the context of the study material?

  It represents the evaluation of the function \( f \) divided by the function \( g \) at \( x = 2 \).
• How do you calculate \( g[f(x)] \) when \( f(x) = 2x + 5 \) and \( g(x) = x + 6 \)?


  You substitute \( f(x) \) into \( g(x) \) to get \( g[f(x)] = (2x + 5) + 6 = 2x + 11 \).
• What are the steps to compute the sum, difference, product, and quotient of two functions?

  1. Identify the functions f(x) and g(x).
  2. For sum: (f + g)(x) = f(x) + g(x).
  3. For difference: (f - g)(x) = f(x) - g(x).
  4. For product: (f ⋅ g)(x) = f(x) · g(x).
  5. For quotient: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0.
• What are the characteristics of a rational expression?


  • Can be written as a ratio of two polynomials
  • Polynomials must have no negative or fractional exponents
  • Variables cannot be inside a radical
• What is an example of a rational expression given in the study material?


  An example is \( \frac{2x^2 + 2x + 3}{5} \).
• What is an example of a non-rational expression given in the study material?


  An example is \( \sqrt{x + 1} \).
• What is a rational equation?

  A rational equation is an equation involving rational expressions that includes an equal sign.
• What is a rational inequality?

  A rational inequality is an inequality involving rational expressions that does not include an equal sign.
• What is the form of a rational function?

  A rational function is of the form \( f(x) = \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomial functions.
• How can a rational equation or inequality be solved?


  It can be solved for all \( x \) values that satisfy the equation or inequality.
• How do you simplify the expressions for the sum, difference, product, and quotient of functions?

  • Combine like terms for sum and difference.
  • Use distributive property for product.
  • Factor and simplify for quotient.
• What is the relationship expressed by a rational function?

  • Expresses a relationship between two variables (e.g., \( x \) and \( y \))
  • Can be represented by a table of values on a graph
• What is an example of a rational function given in the study material?


  An example is \( f(x) = \frac{x^2 + 2x + 3}{x + 1} \).
• What is the date mentioned in the study material for the rational functions section?

  The date is September 9.
• What is the date mentioned in the study material for the composite functions section?

  The date is September 5.