Functions Review

Inverse Functions

Find the inverse of f(x) = 2x + 3 by swapping x and y and solving, then verify the composition f(f⁻¹(x)) = x.

Verify: f(f⁻¹(x)) = m · ((x - b)/m) + b = (x - b) + b = x. Example: f(x) = 2x + 3. Swap: x = 2y + 3. Solve: y = (x - 3)/2. So f⁻¹(x) = (x - 3)/2. Check: 2·((x-3)/2)+3 = x.

Example

Swap x and y, solve for y, and verify the inverse composition.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y=2x+3y= 2x + 3

Step 2 — Swap x and y

Swap x and y: x = 2y + 3.

x=2y+3x= \hlmath{2y} + \hl{3}

Step 3 — Subtract constant

Subtract 3 from both sides, leaving 2y.

x−3=2yx- \hl{3} = 2y

Step 4 — Divide by coefficient

Divide by 2: y = (x - 3) / 2.

y=12(x−3)y= \hlmath{\frac{1}{2}} (x- \hl{3} )

Step 5 — Inverse

The inverse is f inverse of x equals (x - 3) / 2.

g(x)=x−32g(x)=\frac{x- \hl{3} }{ \hl{2} }

Step 6 — Verify

Verify by composing: 2 times (x - 3) / 2, plus 3, simplifies to x.

2⋅x−32+3=x2 \cdot\frac{x- 3 }{ 2 }+ 3 =x
inverse-functions To find the inverse of a linear function f(x) = mx + b: Step 1: Write y = mx + b. Step 2: Swap x and y: x = my + b. Step 3: Solve for y: y = (x - b) / m. Step 4: Write f⁻¹(x) = (x - b) / m.