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OCR GCSE Maths (Higher) Study Notes

2.11.2 Inverse Functions

Inverse functions are a fundamental concept in algebra that allow us to find a function that reverses the effect of another function. This means if we have a function f(x)f(x) that takes an input xx and produces an output yy, the inverse function, denoted as f(x)f(x), will take yy as an input and produce the original xxas an output. Understanding how to find and use inverse functions is crucial for solving equations and understanding the relationship between variables in various mathematical contexts.

Understanding Inverse Functions

To understand inverse functions, one must grasp that the inverse essentially "undoes" the action of the original function. For a function to have an inverse, each input must have a unique output, and each output must come from a unique input. This property is known as being 'one-to-one'. A graphical representation of a function that has an inverse is a curve that passes the horizontal line test - meaning no horizontal line intersects the graph more than once.

Inverse Functions illustration

Image courtesy of Wikimedia

Characteristics of Inverse Functions

  • One-to-one correspondence: For f(x)f(x) to have an inverse, no two different inputs can map to the same output.
  • Function notation: The inverse of f(x)f(x) is denoted as f(x)f(x), which is read as "f inverse of x".
  • Domain and Range: The domain of f(x)f(x) becomes the range of f(x)f(x), and vice versa.
  • Graphical relationship: The graph of f(x)f(x) is a reflection of the graph of f(x)f(x) across the line y=x. y = x.
Inverse Function Mapping

Image courtesy of Mathematics Monster

Finding Inverse Functions

To find the inverse of a function, follow these steps:

1. Replace f(x)f(x) with yy.

2. Swap xx and yy.

3. Solve for yy, which becomes f(x)f(x).

Example 1: Linear Function

Given f(x)=2x+3f(x) = 2x + 3, find f(x)f(x).

1. Replace f(x)f(x) with yy: y=2x+3y = 2x + 3.

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

3. Solve for y:y=x32y: y = \frac{x - 3}{2}

Hence, f1(x)=x32.f^{-1}(x) = \frac{x - 3}{2}.

Example 2: Quadratic Function

Consider f(x)=x2+4x+3f(x) = x^2 + 4x + 3, where x2x \geq -2to ensure the function is one-to-one.

1. Replace f(x)f(x) with yy: y=x2+4x+3y = x^2 + 4x + 3.

2. Swap xxand yy: x=y2+4y+3x = y^2 + 4y + 3.

3. Solve for yy: This step involves completing the square or using the quadratic formula. After manipulation, you find that yy in terms of xx corresponds to the inverse function, ensuring to only consider the branch where x2x \geq -2.

Composite Functions and Inverse Functions

Understanding composite functions is key to working with inverses. The composition of a function ffwith its inverse f1f^{-1} will always yield the original input value for xx, i.e., f(f1(x))=xf(f^{-1}(x)) = xand f1(f(x))=xf^{-1}(f(x)) = x.

Example 3: Verifying Inverses

If f(x)=3x5f(x) = 3x - 5 and g(x)=x+53g(x) = \frac{x + 5}{3}, show that ff and gg are inverses.

1. Compute f(g(x))f(g(x)): f(g(x)f(g(x) = 3(x+53)5=x3\left(\frac{x + 5}{3}\right) - 5 = x.

2. Compute g(f(x))g(f(x)): g(f(x)g(f(x)= (3x5)+53=x\frac{(3x - 5) + 5}{3} = x.

Since both compositions return xx, ff and gg are indeed inverses of each other.

Practice Questions

1. Find the inverse function of f(x)=5x7f(x) = 5x - 7.

2. Given f(x)=x+3f(x) = \sqrt{x + 3}, find f1(x)f^{-1}(x).

3. If g(x)=12x4,g(x) = \frac{1}{2x - 4}, determine (g1(x)(g^{-1}(x).

Solutions

1. For f(x)=5x7f(x) = 5x - 7, f1(x)=x+75f^{-1}(x) = \frac{x + 7}{5}.

2. For f(x)=x+3f(x) = \sqrt{x + 3}, swapping xx and yy and solving for yy gives f1(x)=x23f^{-1}(x) = x^2 - 3, considering the domain x0x \geq 0.

3. For g(x)=12x4g(x) = \frac{1}{2x - 4}, g1xg^{-1}x= 1+4x2x\frac{1 + 4x}{2x}, ensuring to follow the steps of swapping xx and yy and solving fory y.

Key Takeaways

  • Inverse functions reverse the operation of the original function.
  • To find the inverse, swap xx and yy in the equation and solve for yy.
  • The graph of an inverse function is the reflection of the original function across the line y=xy = x.
  • Verifying inverses can be done by showing that f(f1(x)=xf(f^{-1}(x) = x and f1(f(x))=xf^{-1}(f(x)) = x.

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