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MathidiaMathematics Academy
Functions, iteration and numerical methods
Lesson 17 of 75
Easy practice

Equations involving functions

Objective. Solve equations written in formal function notation.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Equations involving functions require solving for an unknown input or parameter using formal notation, such as finding x when f(x) equals a given value, or finding a constant k when f(k)=f(2k). The key step is substituting the unknown carefully into the function's rule exactly as defined, then solving the resulting equation using standard algebraic techniques. When two function expressions are set equal, such as f(x)=g(x), this is solved the same way as any other equation after full substitution. For example, if f(x)=x²−5, solving f(x)=4 means x²−5=4, so x²=9 and x=±3.

Worked example

Given f(x) = 2x²−3, solve the equation f(x) = 15.

Given f(x) = 2x²−3
Illustrated card showing Given f(x) = 2x²−3
  1. 1Substitute into the function: 2x²−3 = 15
  2. 2Add 3 to both sides: 2x² = 18
  3. 3Divide by 2: x² = 9
  4. 4Take square roots: x = ±3

Answer: x = 3 or x = −3

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Common mistake. A common mistake is substituting the output value into x by mistake, solving f(x)=10 for f(x)=2x+1 by writing 2(10)+1 instead of setting 2x+1=10 and solving for x; the given value is what f(x) equals, not x itself.

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