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Functions Flashcards

All 11 cards in this deck

What is a function?

A rule that assigns exactly one output to each input value of xx.
e.g. f(x)=x2−9f(x) = x^2 - 9

What is the domain of a function?

The set of input values xx that the function is defined for.
e.g. f(x)=1xf(x)=\frac{1}{x} has domain all xx except x=0x=0

What are the steps to evaluate a function at a given value?
e.g. f(x)=x2−9f(x) = x^2 - 9, work out f(4)f(4)

  1. Replace every xx with the value: 42−94^2 - 9
  2. Work it out: 16−9=716 - 9 = 7

What are the steps to solve f(x)=kf(x) = k for xx?
e.g. f(x)=2x+1f(x) = 2x + 1, solve f(x)=−5f(x) = -5

  1. Set the expression equal to the value: 2x+1=−52x + 1 = -5
  2. Solve the equation: 2x=−62x = -6, so x=−3x = -3

What are the steps to work out the range of a function over a given domain?
e.g. f(x)=x2+1f(x) = x^2 + 1 for 1≤x≤71 \le x \le 7

  1. Substitute the endpoints of the domain: f(1)=2f(1) = 2, f(7)=50f(7) = 50
  2. Check if a turning point lies inside the domain (x=0x = 0 is not in 1≤x≤71 \le x \le 7)
  3. State the range, smallest value first: 2≤f(x)≤502 \le f(x) \le 50

True or false? fg(x)fg(x) means apply ff first and then apply gg.

False. fg(x)=f(g(x))fg(x) = f(g(x)) — the inner function gg is applied first.

What are the steps to evaluate a composite function at a number?
e.g. f(x)=2x+1f(x) = 2x + 1, g(x)=x2g(x) = x^2, work out fg(3)fg(3)

  1. Work out the inner function: g(3)=9g(3) = 9
  2. Substitute that into the outer function: f(9)=2×9+1=19f(9) = 2 \times 9 + 1 = 19

What are the steps to form and simplify an expression for fg(x)fg(x)?
e.g. f(x)=2x+1f(x) = 2x + 1, g(x)=x2g(x) = x^2

  1. Write fg(x)=f(g(x))fg(x) = f(g(x))
  2. Replace xx in ff by g(x)g(x): 2(x2)+12(x^2) + 1
  3. Expand and simplify: fg(x)=2x2+1fg(x) = 2x^2 + 1

What are the steps to solve an equation involving a composite function?
e.g. f(x)=2x+1f(x) = 2x + 1, g(x)=x2g(x) = x^2, solve fg(x)=19fg(x) = 19

  1. Form the composite: fg(x)=2x2+1fg(x) = 2x^2 + 1
  2. Set it equal to the value: 2x2+1=192x^2 + 1 = 19
  3. Solve: x2=9x^2 = 9, so x=3x = 3 or x=−3x = -3

What are the steps to find the inverse function f−1(x)f^{-1}(x)?
e.g. f(x)=2−x3f(x) = \frac{2-x}{3}

  1. Write y=2−x3y = \frac{2-x}{3}
  2. Rearrange to make xx the subject: 3y=2−x3y = 2 - x, so x=2−3yx = 2 - 3y
  3. Replace yy with xx: f−1(x)=2−3xf^{-1}(x) = 2 - 3x

What are the steps to write f−1(x)+gf(x)f^{-1}(x) + gf(x) as a single simplified expression?
e.g. f(x)=x+1f(x) = x + 1, g(x)=x2g(x) = x^2

  1. Find the inverse: f−1(x)=x−1f^{-1}(x) = x - 1
  2. Form the composite: gf(x)=(x+1)2=x2+2x+1gf(x) = (x+1)^2 = x^2 + 2x + 1
  3. Add and collect terms: x2+3xx^2 + 3x