Function notation, composition and transformations
f(3) asks for an output. f(x) = 7 asks for an input. Almost every mistake in this topic is answering one of those when the stem asked the other.
The function f is defined by f(x) = 4x − 5, and f(a) = 27 for some constant a. What is the value of f(a + 2)?
f(x) = 4x − 5, f(a) = 27
First run the question backwards: 4a − 5 = 27 gives 4a = 32 and a = 8. Then run it forwards at the new input: f(10) = 4(10) − 5 = 35. (Quicker still: adding 2 to the input adds 4 · 2 = 8 to the output, and 27 + 8 = 35.) A answers with a itself, the value of the input rather than the output asked for; B answers with a + 2, which is only the new input; C adds 2 to the output instead of to the input — but f(x) climbs 4 for every 1 you add to x, not 1.
1
f(3) means: put 3 wherever the x is.
The bracket is a slot, not a multiplication. Whatever sits in it replaces every x in the rule.
f(x) = 4x − 5
the rule
f(3) = 4(3) − 5 = 7
a number in the slot
f(−2) = 4(−2) − 5 = −13
keep the brackets and the sign survives
f(a + 1) = 4(a + 1) − 5 = 4a − 1
a whole expression in the slot
A table hands you the same information as a list. A graph hands it to you as a picture: go to x = 4 on the horizontal axis, travel to the curve, and read the height.
x
0
2
4
6
8
f(x)
11
5
3
5
11
f(x + 1) is not f(x) + 1
For f(x) = 4x − 5, f(x + 1) = 4x − 1 while f(x) + 1 = 4x − 4. The first
changes what goes in; the second changes what comes out. The whole of the
transformation section later in this lesson rests on that difference.
2
“Solve f(x) = 7” is the same fact read backwards.
You are handed the output and asked for the input. Set the rule equal to the number and solve for x.
4x − 5 = 7
the rule, set equal to the output
4x = 12 → x = 3
and f(3) = 7, from step 1
4x − 5 = 19 → x = 6
same move, different output
Read the two lines together: “f(3) = 7” and “the solution of f(x) = 7 is
x = 3” say exactly the same thing. One names the height, the other names
the place.
The stem asks
You are given
You report
“what is f(3)”
the input 3
an output
“what is the solution to f(x) = 7”
the output 7
an input
“for what value of x is f(x) = 0”
the output 0
an x-intercept
On a graph the reverse question is a horizontal line. Draw y = 5 across
the picture and every crossing point gives one solution — here two of
them, x = 2 and x = 6. The table says the same thing: 5 appears twice in
the f(x) row. An input question has one answer; an output question can
have none, one or several.
3
f(g(x)) runs the inner machine first.
Work outward from the innermost bracket, exactly as you would with arithmetic.
start1g →−2f →1
f(x) = 2x + 5, g(x) = x² − 3
two rules
g(1) = 1 − 3 = −2
inner bracket first
f(g(1)) = f(−2) = −4 + 5 = 1
feed the result into f
Swap the order and the answer changes.
f(1) = 2 + 5 = 7
now f is the inner one
g(f(1)) = g(7) = 49 − 3 = 46
1, not 46 — the order is the whole question
f(g(1)) = 1 · g(f(1)) = 46
The wrong order is always one of the choices
Whenever an item asks for f(g(3)), the value of g(f(3)) is waiting in the
answer list. Circle the inner function before you compute anything. If
the stem gives a table instead of formulas, find the inner value in the
x column first, then take that result back to the x column again.
f(g(x)) = 2(x² − 3) + 5 = 2x² − 1
g's whole rule into f's slot
g(f(x)) = (2x + 5)² − 3 = 4x² + 20x + 22
the middle term 2(2x)(5) = 20x is not optional
4
Inside the bracket moves it sideways; outside moves it up.
A change outside the bracket does what it says. A change inside the bracket does the opposite of what it says.
New function
What happens to the graph
Why
f(x) + k
up k units
every output is raised by k
f(x) − k
down k units
outside is honest
f(x + h)
left h units
the input reaches its value sooner
f(x − h)
right h units
the input has to catch up
−f(x)
reflected over the x-axis
the outputs change sign
f(−x)
reflected over the y-axis
the inputs change sign
a f(x), a > 1
stretched vertically, a times as tall
every height is multiplied by a
a f(x), 0 < a < 1
squashed toward the x-axis
same rule, a smaller multiplier
The sideways rule is the only one that surprises people. (x − 3) needs
x = 3 to produce what x = 0 produced before, so the picture slides toward
the positive direction — the minus sign inside moves the graph right.
Anything outside the bracket behaves normally.
5
Turn the SAT's sentence into a movement.
The stem almost always arrives in one shape: “the function g is defined by g(x) = f(x − 2) + 5”. Read it in two halves.
f(x − 2)2 units right+ 55 units up
g(x) = f(x − 2) + 5
the sentence, as an equation
every point (p, q) on f moves to (p + 2, q + 5)
right 2, up 5
if (1, 4) is on f, then (3, 9) is on g
check: g(3) = f(3 − 2) + 5 = f(1) + 5 = 9 ✓
Substituting the x that makes the bracket simple is the reliable move. To
use f(1), set x − 2 = 1, so x = 3. Never guess which coordinate shifts.
The sign inside is read backwards, every time
f(x − 3) is three units right, not left, and f(x + 4) is
four units left. Both wrong readings sit in the answer
choices of almost every transformation item, which is why it pays to test
one point rather than trust the reflex.
✓
Check: substitute one point back in.
Every claim in this lesson can be verified with a single input.
the openerf(8) = 32 − 5 = 27a = 8 really does give 27 ✓
and forwardsf(10) = 40 − 5 = 35the quantity actually asked for ✓
the reverse questionf(3) = 7, and 4(3) − 5 = 7the same fact both ways ✓
compositionf(g(x)) = 2x² − 1 at x = 1 gives 1matches f(g(1)) = 1 ✓
the shiftg(1) = f(1 − 3) − 3 = f(−2) − 3 = 4 − 3 = 1the gold peak really is at (1, 1) ✓
Remember the flow
Read the bracket as a slot · Output asked? substitute · Input asked? solve · Inner function first · Inside = sideways, reversed
f(3) and “solve f(x) = 3” are opposite questions wearing the same clothes.
Now you try
Practice problem
The function h is defined by h(x) = f(x + 4) − 6. In the xy-plane, the graph of y = h(x) passes through the point (−1, 2). Which of the following points must lie on the graph of y = f(x)?
h(x) = f(x + 4) − 6
Put x = −1 into the definition: h(−1) = f(−1 + 4) − 6 = f(3) − 6. That value is 2, so f(3) − 6 = 2 and f(3) = 8, which puts (3, 8) on the graph of f. A and B use −1 − 4 = −5, moving the input the wrong way; the bracket already contains x + 4, so you evaluate it, you do not undo it. A and C subtract the 6 again instead of adding it back — going from h to f reverses the −6. Read it as a movement if you prefer: h is f shifted 4 left and 6 down, so the point (−1, 2) on h came from (−1 + 4, 2 + 6) = (3, 8) on f.
Hint: substitute the x you were given straight into the
definition and see which f-value appears. The bracket tells you the input to f;
the number outside tells you how to get back to f's output.
