Transforming a function · SAT Math

In g(x)=Af(x−h)+k, the inner expression chooses an input for f. Multiplication by A and addition of k then change the output. Label these two stages instead of treating every visible minus sign as a movement in the same direction. A table supplies old outputs, so only transformations that use listed inputs can be evaluated from it.

Use a table deliberately

Suppose f(3)=8 and g(x)=−2f(x)+5. Then g(3)=−16+5=−11. The negative multiplier reflects the outputs across the horizontal axis before the shift. By contrast, −2[f(x)+5] gives −26 at three, because its shift occurs before multiplication. Parentheses encode a different sequence of operations.

Move a zero by solving an input condition

If f has its only zero at five and g(x)=f(x−2), then g vanishes when x−2=5, so x=7. The graph moves right by two. If the definition were f(x+2), the zero would move to three. Solving the inner equation is more reliable than memorizing the written sign.

Undo a composition in stages

Let f(t)=2t+3 and g(u)=u²+5, with g(f(t))=54 and f(t)>0. Undo the outer shift first: f(t)²=49. The sign condition selects f(t)=7, so 2t+3=7 gives t=2. Taking the negative root would give t=−5, which violates f(t)>0. A composition requires both an order of inversion and a branch decision; replacing f(t) by a bare unknown too early can hide the condition.

Transfer and check

Change the composed output to 74 while keeping f(t)>0. Then f(t)²=69, so the permitted input is t=(√69−3)/2. Verify that f(t)=√69 is positive before squaring it. Separately, a package costs $18 for four hours, then $5 per extra hour: six hours cost$28, while exactly four hours still cost$18. Select the proper branch of each rule before computing.

WORKED EXAMPLE

The table gives values of f. If g(x)=−2f(x)+5, what is g(3)?

$Function values. x: 1, f(x): 1; x: 3, f(x): 5; x: 5, f(x): 9$ Open full-size diagram ↗

Read f(3)=5 from the table.
Apply the stated output transformation: g(3)=−2(5)+5.
The result is -5; the input remains 3 throughout.

Answer: -5

Compare the other answer choices

Omit the output scale factor.

Omit the negative sign on the scale factor.

-20

Apply the output shift before the stretch.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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