A reliable method
For g(f(x)), calculate f(x) before using g. For a piecewise rule, choose the branch using the original input, not the resulting output.
The common trap
Do not multiply f and g when the notation calls for composition. Inverting ax+b requires subtracting b before dividing by a.
Make the connection
An arithmetic sequence is a linear rule indexed by position. Its first term belongs to n=1, so n−1 counts the steps after that first term.
Worked example: choose the requested quantity
f(x) = 2x + 3 for x < 0 and f(x) = x² − 3 for x ≥ 0. What is f(−4)? The input is negative, so use the first branch. Substitute the negative input into the linear rule. The requested value is -5.
Contrast and diagnose an error
f(x) = 3x − 3 and g(x) = x + 4. What is g(f(8))? Apply f to the given input first. Add the constant from g to that output; do not multiply the rules. The requested value is 25. A tempting answer is 17: Use subtraction in the outer rule.
Transfer practice and check
f(x) = 3x + 4. If f(t) = 28, what is t? Try it before reading on. Check: 8. Subtract the output offset from both sides. Divide by the nonzero multiplier to reverse the function.
f(x) = 2x + 3 for x < 0 and f(x) = x² − 3 for x ≥ 0. What is f(−4)?
- The input is negative, so use the first branch.
- Substitute the negative input into the linear rule.
- The requested value is -5.
Compare the other answer choices
Use the nonnegative-input branch.
Subtract the branch constant instead of adding.
Square the negative input with the wrong sign and use the wrong branch.
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.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content