Functions
Composite of Inverses vs Inverse of Composite
In this section, we are focusing on the inverse of the composite function g(f(x)) that is, [g(f(x))]⁻¹.
Recall from earlier that:
g(f(x)) means: apply f first, then apply g to the result.
Now we want to answer a specific question:
Question: When finding the inverse of g(f(x)), do we apply the inverses in the same order as the original functions (g⁻¹ ∘ f⁻¹), or in reverse order (f⁻¹ ∘ g⁻¹)?
Let us investigate this question using the same functions we have been working with:
f(x) = 2x + 1 (adds 1, then multiplies by 2)
g(x) = 3x - 2 (multiplies by 3, then subtracts 2)
From earlier, we already found the inverses of these functions:
(undoes: divides by 2, then subtracts 1?) Wait, no — f⁻¹ subtracts 1 first, then divides by 2
(undoes: adds 2, then divides by 3)
What we want to find out:
- Option A: Apply the inverses in the same order as the original functions → g⁻¹ ∘ f⁻¹
- Option B: Apply the inverses in the reverse order → f⁻¹ ∘ g⁻¹
Let us calculate both options and see which one matches the inverse of the composite we found earlier.
Step 1: Try Option A (same order) → g⁻¹ ∘ f⁻¹
(g⁻¹ ∘ f⁻¹)(x) means: apply f⁻¹ first, then apply g⁻¹ to the result.
(g⁻¹ ∘ f⁻¹)(x) = g⁻¹(f⁻¹(x)) = g⁻¹((x - 1)⁄2)
Step 2: Try Option B (reverse order) → f⁻¹ ∘ g⁻¹
(f⁻¹ ∘ g⁻¹)(x) means: apply g⁻¹ first, then apply f⁻¹ to the result.
(f⁻¹ ∘ g⁻¹)(x) = f⁻¹(g⁻¹(x)) = f⁻¹((x + 2)⁄3)
Step 3: Recall the inverse of the composite function we found earlier
Conclusion — Answering Our Question:
• Option A (same order): g⁻¹ ∘ f⁻¹ = (x + 3)⁄6
• Option B (reverse order): f⁻¹ ∘ g⁻¹ = (x - 1)⁄6
• The actual inverse [g(f(x))]⁻¹ = (x - 1)⁄6
Therefore:
• g⁻¹ ∘ f⁻¹ ≠ [g(f(x))]⁻¹ (applying inverses in the same order gives the WRONG result)
• f⁻¹ ∘ g⁻¹ = [g(f(x))]⁻¹ (applying inverses in reverse order gives the CORRECT result)
Conclusion: The inverse of a composite function is the composite of the inverses in REVERSE order.
In symbols: [g(f(x))]⁻¹ = f⁻¹ ∘ g⁻¹
Proving the Inverse Composite Rule
Prove that for any two functions f and g, [g(f(x))]⁻¹ = f⁻¹(g⁻¹(x)).
Proof:
Let h(x) = g(f(x)). Then h⁻¹(x) is the inverse of h.
By definition of inverse functions: h(h⁻¹(x)) = x and h⁻¹(h(x)) = x.
Consider (f⁻¹ ∘ g⁻¹)(g ∘ f)(x):
(f⁻¹ ∘ g⁻¹)(g(f(x))) = f⁻¹(g⁻¹(g(f(x)))) = f⁻¹(f(x)) = x
Therefore, f⁻¹ ∘ g⁻¹ is the inverse of g ∘ f.
Hence, [g(f(x))]⁻¹ = f⁻¹(g⁻¹(x))
Key Rule for Inverse of Composite:
• [f ∘ g]⁻¹ = g⁻¹ ∘ f⁻¹
• The inverse of a composite is the composite of inverses in reverse order
• This is often remembered as: "The inverse of a composite is the composite of the inverses, but in reverse order."
Summary Table: Functions
| Concept | Description | Example |
|---|---|---|
| Function | Each input has exactly one output | f(x) = 2x + 3 |
| Function Notation | f(x) represents the output for input x | f(2) = 2(2) + 3 = 7 |
| Linear Function | Graph is a straight line: f(x) = mx + c | f(x) = 2x - 1 |
| Gradient (m) | How steep the line is: (y₂ - y₁) ÷ (x₂ - x₁) | m = (10-2) ÷ (4-0) = 2 |
| y-intercept (c) | Where the line crosses the y-axis | At (0, 2), so c = 2 |
| Vertical Line Test | If a vertical line touches the graph more than once, it is not a function | A circle fails the vertical line test |
| Domain | Set of all possible input values (x) | {x | -3 ≤ x ≤ 3} |
| Range | Set of all possible output values (f(x)) | {y | -5 ≤ y ≤ 7} |
| Inverse Function | Reverses the operation of the original function | f⁻¹(x) = (x - 3)/2 |
| Composite Function | Apply one function to the result of another: (f ∘ g)(x) = f(g(x)) | (f ∘ g)(x) = 2(3x) + 1 = 6x + 1 |
Key Takeaways:
• A function gives exactly one output for each input. Use the vertical line test to check.
• Function notation f(x) replaces y in an equation.
• Linear functions have the form f(x) = mx + c, where m is the gradient and c is the y-intercept.
• To evaluate a function, substitute the input value into the function.
• To find the function from a graph, calculate the gradient and identify the y-intercept.
• The domain is the set of all inputs. The range is the set of all outputs.
• To find an inverse, swap x and y and solve for y.
• Composite functions apply one function to the result of another. The order matters.
NB: Functions are a fundamental concept in mathematics. They describe how one quantity depends on another. Linear functions are the simplest type of function, but they are very useful for modelling real-world situations like cost, distance, and speed. Make sure you can evaluate functions, draw their graphs, find the function from a graph, find inverses, and work with composite functions.