GED Mathematical Reasoning
Function Notation
f(x) is a machine: you feed a number in, it does something, a number comes out. The notation looks intimidating but f(3) simply means "put 3 in wherever x appears." It does NOT mean f times 3.
The words first
- function
- a rule that turns each input into exactly one output
- f(x)
- read aloud as "f of x" — the output when you feed in x
- input / output
- what you put in / what comes out
- evaluate
- actually do the arithmetic for a specific input
The method
WHAT THE NOTATION MEANS
f(x) = 2x + 3
f(4) means put 4 where x is
f(4) = 2(4) + 3 = 8 + 3 = 11
f(4) is NOT f × 4.
READING IT FROM A TABLE
x | 1 2 3
f(x) | 5 8 11
f(2) = 8 — go to x = 2, read the output
GOING BACKWARDS
"f(x) = 11, what is x?" → find 11 in the output row, read off x = 3
PUTTING IN SOMETHING THAT IS NOT A NUMBER
f(a + 1) = 2(a + 1) + 3 = 2a + 2 + 3 = 2a + 5
Worked all the way through
The problem
If f(x) = x² − 4x + 1, find f(−2).
- Step 1 — Rewrite the rule with empty slots where x was.f( ) = ( )² − 4( ) + 1
- Step 2 — Drop −2 into every slot. Keep the parentheses.f(−2) = (−2)² − 4(−2) + 1
- Step 3 — Do the exponent first.(−2)² means −2 times −2, which is +4. Negative times negative is positive.
- Step 4 — Do the multiplication.−4 × (−2) = +8. Again, two negatives make a positive.
- Step 5 — Add it all up left to right.4 + 8 + 1 = 13
On the board
f(x) = x² − 4x + 1
the rule
f(−2) = (−2)² − 4(−2) + 1
put −2 in every slot
= 4 − 4(−2) + 1
(−2)² = −2 × −2 = +4
= 4 + 8 + 1
−4 × −2 = +8
= 13
Answer
13
A second one, in a different form
If f(x) = 3x − 5, find f(4).
f(x) = 3x − 5
the rule
f(4) = 3(4) − 5
put 4 where x is
= 12 − 5
= 7
Answer
7
The usual mistake
Reading f(4) as f times 4.
Check yourself
Those parentheses around the −2 are doing real work. Without them you would write −2² and get −4 instead of +4, and the whole answer would be wrong.
Practice it
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