Functions Worksheets

Evaluate functions, compose two functions and find the inverse of a linear function. Grades 11–12.

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What these worksheets practise

A function is a rule that turns each input into exactly one output. Writing f(x)=2x2−3x+1f(x) = 2x^{2} - 3x + 1 names the rule ff, and f(−2)f(-2) means "the output of ff when the input is −2-2". Pre-Calculus treats functions as objects of their own: they can be chained together (composition) and undone (inverses).

  • Easy — evaluate a quadratic function f(x)=ax2+bx+cf(x) = ax^{2} + bx + c at a whole number, often a negative one.
  • Medium — composition: with f(x)=ax+bf(x) = ax + b and g(x)=x2+cg(x) = x^{2} + c, find (f∘g)(k)(f \circ g)(k).
  • Hard — find the inverse f−1(x)f^{-1}(x) of a linear function f(x)=ax+bf(x) = ax + b.

Worked examples

Easy: f(x)=2x2−3x+1f(x) = 2x^{2} - 3x + 1, find f(−2)f(-2)

Replace every xx with (−2)(-2), keeping the parentheses:

f(−2)=2(−2)2−3(−2)+1=2(4)+6+1=15f(-2) = 2(-2)^{2} - 3(-2) + 1 = 2(4) + 6 + 1 = 15

Medium: f(x)=3x−2f(x) = 3x - 2, g(x)=x2+1g(x) = x^{2} + 1, find (f∘g)(2)(f \circ g)(2)

(f∘g)(2)(f \circ g)(2) means f(g(2))f(g(2)): apply gg first, then ff.

g(2)=22+1=5f(5)=3(5)−2=13g(2) = 2^{2} + 1 = 5 \qquad f(5) = 3(5) - 2 = 13

Answer: (f∘g)(2)=13(f \circ g)(2) = 13.

Hard: f(x)=4x+5f(x) = 4x + 5, find f−1(x)f^{-1}(x)

Write y=4x+5y = 4x + 5, swap xx and yy, and solve for yy:

x=4y+5⇒x−5=4y⇒y=x−54x = 4y + 5 \quad\Rightarrow\quad x - 5 = 4y \quad\Rightarrow\quad y = \frac{x - 5}{4}

Answer: f−1(x)=x−54f^{-1}(x) = \dfrac{x - 5}{4}.

Check your answers

An inverse undoes the function, so f(f−1(x))f(f^{-1}(x)) must give back xx. With x=9x = 9: f−1(9)=9−54=1f^{-1}(9) = \tfrac{9 - 5}{4} = 1 and f(1)=4+5=9f(1) = 4 + 5 = 9.

Common mistakes

  • Losing the sign of a negative input. 2(−2)2=2(4)=82(-2)^{2} = 2(4) = 8, not −8-8. Always substitute with parentheses.
  • Composing in the wrong order. (f∘g)(2)=f(g(2))=13(f \circ g)(2) = f(g(2)) = 13, but (g∘f)(2)=g(f(2))=g(4)=17(g \circ f)(2) = g(f(2)) = g(4) = 17. The function written on the right is applied first.
  • Confusing the inverse with a reciprocal. f−1(x)f^{-1}(x) is not 14x+5\dfrac{1}{4x + 5}. The −1-1 means "undo", not "divide into 1".
  • Dividing only part of the expression. x−54\dfrac{x - 5}{4} is not x4−5\dfrac{x}{4} - 5.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. f(x) = −x² + 2x − 1, find f(3)
  2. f(x) = −3x² − 3x − 2, find f(3)
  3. f(x) = 3x² − x − 4, find f(3)

Medium

  1. f(x) = 4x + 4, g(x) = x²
    Find (f∘g)(−3)
  2. f(x) = 3x − 3, g(x) = x² − 1
    Find (f∘g)(−3)
  3. f(x) = 2x − 2, g(x) = x² + 3
    Find (f∘g)(3)

Hard

  1. f(x) = 5x + 7, find f−1(x)
  2. f(x) = 5x + 3, find f−1(x)
  3. f(x) = 5x + 4, find f−1(x)
Show answers
  1. f(3) = -4
  2. f(3) = -38
  3. f(3) = 20
  4. (f∘g)(−3) = 40
  5. (f∘g)(−3) = 21
  6. (f∘g)(3) = 22
  7. f⁻¹(x) = (x − 7) / 5
  8. f⁻¹(x) = (x − 3) / 5
  9. f⁻¹(x) = (x − 4) / 5

Tips for teachers and parents

  • The "function machine" picture works well for composition: the output of the first machine is fed into the second.
  • On Easy worksheets, ask students to write the substitution with parentheses before simplifying; most errors come from skipping that line.
  • For inverses, have students check one value with f(f−1(x))=xf(f^{-1}(x)) = x. It shows immediately whether the inverse is right.

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