Gen math

• Evaluate a function


  Replace the variable in the function with a value from the function's domain and compute the result
• Cannot evaluate a function at a value not in its domain
• Domain of a function


  The set of values of x for which the function is defined
• When temperature rises to 30C, the top speed is reduced to 10 km per hour
• V(0)

  Initial velocity of the ball is 20 m/s
• V(1)


  After 1 second, the ball is traveling more slowly, at 10.2 m/s
• f(x)

  x^2 + 3x - 1
• g(x)

  2x^2 - 3x + 5
• h(x)

  4x - 2
• k(x)

  1/(x-1)
• Given the function f(x) = (x^2 + 3x - 1)/(x^2 - 1), find the values of

  1. f(2)

  2. f(-1)

  3. f(0)

  4. f(1)

  5. f(3)
• Is f(-1) the same as f(1)?
• A computer shop charges P20.00 per hour (or a fraction of an hour) for the first two hours and an additional P10.00 per hour for each succeeding hour. Find how much you would pay if you used one of their computers for

  1. 40 minutes

  2. 3 hours

  3. 150 minutes
• Under certain circumstances, a rumor spreads according to the equation p(t) = 1 / (1 + 2e^(-t/4)), where p(t) is the proportion of the population that knows the rumor (t) days after the rumor started. Find

  1. p(4)

  2. p(10)
• The proportion of the population that knows the rumor after 4 days and 10 days
• Operations on algebraic expressions
• Least common denominator (LCD)


  The denominator of the resulting fraction when adding or subtracting fractions
• Steps to add or subtract fractions

  1. Find the LCD

  2. Rewrite the fractions as equivalent fractions with the same LCD

  3. The LCD is the denominator of the resulting fraction

  4. The sum or difference of the numerators is the numerator of the resulting fraction
• Steps to multiply fractions or rational expressions

  1. Rewrite the numerator and denominator in terms of its prime factors

  2. Common factors in the numerator and denominator can be simplified as "1" (cancelling)

  3. Multiply the numerators together to get the new numerator

  4. Multiply the denominators together to get the new denominator
• Steps to divide fractions or rational expressions

  Multiply the dividend with the reciprocal of the divisor
• Sum of functions f and g

  f(x) + g(x)
• Difference of functions f and g

  f(x) - g(x)
• Product of functions f and g

  f(x)g(x)
• Quotient of functions f and g

  f(x)/g(x), excluding the values of x where g(x) = 0
• Rational


  (in classical economic theory) economic agents are able to consider the outcome of their choices and recognise the net benefits of each one
• Rational agents will select the choice which presents the highest benefits
• Consumers act rationally by

  Maximising their utility
• Producers act rationally by


  Selling goods/services in a way that maximises their profits
• Workers act rationally by

  Balancing welfare at work with consideration of both pay and benefits
• Governments act rationally by

  Placing the interests of the people they serve first in order to maximise their welfare
• Rationality in classical economic theory is a flawed assumption as people usually don't act rationally
• Demand curve shifting right

  Increases the equilibrium price and quantity
• Marginal utility

  The additional utility (satisfaction) gained from the consumption of an additional product
• If you add up marginal utility for each unit you get total utility
• Polynomial function


  A function that can be written in the form p(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a positive integer
• Rational function


  A function of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomial functions and q(x) is not the zero function
• The domain of a rational function f(x) = p(x)/q(x) is the set of all values of x where q(x) is not zero
• Solving a rational equation


  Eliminate denominators by multiplying each term by the least common denominator

  2. Check for extraneous solutions
• Solving a rational inequality


  Rewrite as a single rational expression on one side and 0 on the other

  2. Locate the x values where the expression is 0 or undefined

  3. Partition the number line into intervals based on the values found in step 2

  4. Test a point in each interval to determine the sign of the expression
• To solve rational inequalities


  1. Rewrite the inequality as a single rational expression on one side of the inequality symbol and 0 on the other side

  2. Determine over what intervals the rational expression takes on positive and negative values

  3. Locate the x values for which the rational expression is zero or undefined

  4. Mark the numbers found in (iii) on a number line

  5. Select a test point within the interior of each interval in (iv)

  6. Summarize the intervals containing the solutions