AMC 10 · 2004 · #13

Grade 11 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

function-compositionsystems-of-equationspolynomial-factoringinverse-function work-backwardsconvert-to-algebra ↑ Prerequisites: function-composition

📏 Medium solution 💡 3 insights

Problem

A linear function has an inverse, and that inverse is the linear function you get by swapping its two numbers. Find the sum of those two numbers.

Pick an answer.

(A)

-2

(B)

-1

(C)

(D)

(E)

How to solve

Strategy Work Backwards

An inverse function is literally the instruction 'undo what f did', so Tool #11 (Work Backwards) computes f⁻¹ straight from f by unwinding the steps multiply-by-a-then-add-b. That produces a formula for f⁻¹ in terms of a and b, and the problem hands us a second formula for the same function. Tool #13 (Convert to Algebra) matches the two formulas coefficient by coefficient, which turns a statement about functions into two ordinary equations. Tool #6 (Guess and Check) closes the loop at the end by testing the resulting function against the definition of an inverse.

1STEP 1

Rule out the flat case

A flat function has no inverse, so the slope cannot be zero.

a=0 → f(x)=b for all x → f not invertible, so a ≠ 0

2STEP 2

Undo f to get the real inverse

Undoing the steps in reverse order gives the real inverse.

y=ax+b → x=(y-b)/a → f⁻¹(x)=1/ax-b/a

3STEP 3

Match the two formulas for the same function

Matching slopes and constants turns the claim into two equations.

1/ax-b/a=bx+a for all x ⟺ b=1/a and a=-b/a

4STEP 4

Solve the pair of equations

Only one real pair survives, both numbers equal to negative one.

b=-a², ab=1 → -a³=1 → a³+1=(a+1)(a²-a+1)=0 → a=-1, b=-1

5STEP 5

Check the function it produces

Applying it twice returns the input, so the sum is -2, choice (A).

f(x)=-x-1, f(f(x))=-(-x-1)-1=x, a+b=-2 (A)

Answer

-2

The picture agrees with the algebra. The graph of f⁻¹ is always the mirror image of the graph of f across the line y=x. Here f(x)=-x-1 has slope -1, so its graph is perpendicular to that mirror line, and a line perpendicular to a mirror reflects onto itself. So f being its own inverse is exactly what the geometry predicts, not a coincidence. The word 'real' in the problem is doing real work: over the complex numbers a³=-1 has three roots and a+b would not be a single number, so the restriction to real a and b is what makes the question well posed. The result a+b=-2 is choice (A).

💡Key takeaway

Build the inverse yourself by undoing f step by step, then set your formula equal to the one the problem gives and match coefficients: the two equations that fall out pin a and b down completely.

Rule out the flat case
Undo f to get the real inverse
Match the two formulas for the same function
Solve the pair of equations
Check the function it produces

Rule out the flat case

Undo f to get the real inverse

Match the two formulas for the same function

Solve the pair of equations

Check the function it produces

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