LEMMA

Functions

Function notation and evaluation

ALEKS placement

f(x) is a machine: replace every x with whatever sits inside the parentheses.

What this covers

  • Rewrite the rule with empty parentheses where each x was.
  • Drop the input into every slot, keeping it in parentheses.
  • Simplify, respecting the order of operations.
  • For an expression as input, expand carefully.
  • For a piecewise rule, choose the branch using the original input before substituting into that branch.

Worked example

Worked example

If f(x) = x² - 4x + 1, find f(-3).

  1. (−3)2−4(−3)+1Substitute -3 into both x slots, keeping the parentheses.
  2. 9+12+1=22(-3)² is 9 and -4(-3) is +12, both positive.
Another worked example

If f(x) = 6x − 2, find f(3).

  1. 6(3)−2Replace x with 3, keeping it in parentheses.
  2. 18−2=16Multiply before subtracting. So f(3) = 16.

A common mistake

A mistake Lemma catches

From (−3)2−4(−3)+1

−9+12+19+12+1

“Substituting x = −3 squares the whole input: (−3)² = (−3)(−3) = 9. Writing −9 squares only the 3 and keeps the minus outside, −3² = −(3²) = −9. Keep the brackets around a negative input.”

Try one

Sample problem

If f(x) = 6x − 2, find f(3).

f(x)=6x−2,f(3)=?

Practice evaluate functions free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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