Question

Algebraically determine the equation of the inverse of each function. a) \(f(x)=7 x\) b) \(f(x)=-3 x+4\) c) \(f(x)=\frac{x+4}{3}\) d) \(f(x)=\frac{x}{3}-5\) e) \(f(x)=5-2 x\) f) \(f(x)=\frac{1}{2}(x+6)\)

Short answer

a) \(f^{-1}(x) = \frac{x}{7}\), b) \(f^{-1}(x) = \frac{4 - x}{3}\), c) \(f^{-1}(x)= 3x - 4\), d) \(f^{-1}(x) = 3x + 15\), e) \(f^{-1}(x)= \frac{5 - x}{2}\), f) \(f^{-1}(x)=2x - 6\)

Step-by-step solution

Understand the Concept of Inverses

An inverse function reverses the effect of the original function. If a function is denoted as \( f(x) \), its inverse is denoted as \( f^{-1}(x) \). The key idea is that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Swap \( x \) and \( y \)

To find the inverse function, start by replacing \( f(x) \) with \( y \). This helps to manipulate the equation easily. Then swap \( x \) and \( y \). You will solve for \( y \) later to get the inverse.

Solve for \( y \) in Each Function

Solve each equation for \( y \) after swapping \( x \) and \( y \).

a) Find the Inverse of \( f(x) = 7x \)

Start with \( y = 7x \). Swap \( x \) and \( y \) to get \( x = 7y \). Solve for \( y \): \[ y = \frac{x}{7} \]. So, the inverse function is \[ f^{-1}(x) = \frac{x}{7} \].

b) Find the Inverse of \( f(x) = -3x + 4 \)

Start with \( y = -3x + 4 \). Swap \( x \) and \( y \) to get \( x = -3y + 4 \). Solve for \( y \): \[ x - 4 = -3y \], then \[ y = \frac{4 - x}{3} \]. So, the inverse function is \[ f^{-1}(x) = \frac{4 - x}{3} \].

c) Find the Inverse of \( f(x) = \frac{x + 4}{3} \)

Start with \( y = \frac{x + 4}{3} \). Swap \( x \) and \( y \) to get \( x = \frac{y + 4}{3} \). Solve for \( y \): \[ 3x = y + 4 \], then \[ y = 3x - 4 \]. So, the inverse function is \[ f^{-1}(x) = 3x - 4 \].

d) Find the Inverse of \( f(x) = \frac{x}{3} - 5 \)

Start with \( y = \frac{x}{3} - 5 \). Swap \( x \) and \( y \) to get \( x = \frac{y}{3} - 5 \). Solve for \( y \): \[ x + 5 = \frac{y}{3} \], then \[ y = 3(x + 5) \] or \[ y = 3x + 15 \]. So, the inverse function is \[ f^{-1}(x) = 3x + 15 \].

e) Find the Inverse of \( f(x) = 5 - 2x \)

Start with \( y = 5 - 2x \). Swap \( x \) and \( y \) to get \( x = 5 - 2y \). Solve for \( y \): \[ 2y = 5 - x \], then \[ y = \frac{5 - x}{2} \]. So, the inverse function is \[ f^{-1}(x) = \frac{5 - x}{2} \].

f) Find the Inverse of \( f(x) = \frac{1}{2}(x + 6) \)

Start with \( y = \frac{1}{2}(x + 6) \). Swap \( x \) and \( y \) to get \( x = \frac{1}{2}(y + 6) \). Solve for \( y \): \[ 2x = y + 6 \], then \[ y = 2x - 6 \]. So, the inverse function is \[ f^{-1}(x) = 2x - 6 \].

Key concepts

function inversion

The concept of a function and its inverse is like having a reversible machine. The function itself can be seen as a 'process' that converts input (x) into output (f(x)). An inverse function essentially reverses this process. If you input the output back into the inverse function, you end up with the original input. Symbolically, if we have a function f(x), the inverse is denoted as f^{-1}(x).
To understand this, imagine you have a function where f(2)=5. The inverse should satisfy f^{-1}(5)=2. This relationship holds true more generally such that passing a value through the function and then through its inverse will yield the original value: f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

• Function: x -> y
• Inverse: y -> x

algebraic manipulation

To find an inverse function, algebraic manipulation plays a vital role. You start by expressing the function equation in a form that is easier to manipulate. For example, if f(x) = 7x, we start by writing it as y = 7x. Subsequently, swap x and y, giving x = 7y. The objective is to solve this new equation for y.
Manipulating this equation algebraically, we get: y = x/7. Hence the inverse function is f^{-1}(x) = x/7.
Strong algebraic manipulation skills help re-arrange and simplify equations, allowing us to isolate variables effectively. Key operations include: addition, subtraction, multiplication, and division of terms.

• Simplify complex expressions
• Isolate variables
• Apply inverse operations

solving equations

Solving equations is a critical step in finding inverses. After swapping the variables, you need to isolate the new variable (usually y). Let's see an example using f(x) = -3x + 4.
Step-by-step, the process looks like this: 1. Substitute y for f(x): y = -3x + 4.
2. Swap x and y: x = -3y + 4.
3. Solve for y: x - 4 = -3y => y = (4 - x)/3.
This means the inverse function is f^{-1}(x) = (4 - x)/3.
Solving involves operations like: shifting terms to the other side, simplifying fractions, and applying inverse operations. These operations help to untangle the equation and find the value we need.

• Isolate desired variable
• Use inverse operations
• Simplify fraction or expression

precalculus

Function inversion lies at the heart of precalculus, helping bridge to more complex calculus concepts. Precalculus covers fundamental mathematical skills needed for calculus, including function behavior and manipulation. Understanding inverse functions prepares students for concepts like derivative and integral functions in calculus.
Take f(x)= (x+4)/3 in this context:
1. Swap x and y: x = (y+4)/3.
2. Solve for y: 3x = y + 4 => y = 3x - 4.
This gives the inverse function f^{-1}(x) = 3x - 4.
Inverses are straightforward examples of how functions can 'undo' each other's operations, emphasizing themes of function manipulation and transformation vital for calculus.

• Function behavior
• Transition to calculus
• Critical mathematical skills

step-by-step solutions

Breaking down the process into clear, manageable steps makes finding inverse functions less daunting. Let's go through a function: f(x) = (x/3) - 5:
1. Start with y = x/3 - 5.
2. Swap x and y: x = y/3 - 5.
3. Solve for y: x + 5 = y/3 => y = 3(x + 5).
Thus, the inverse function is f^{-1}(x) = 3(x + 5).
Each step simplifies the problem, allowing focus on one aspect of the equation at a time. This methodical approach enhances understanding and accuracy, making math problems more accessible.

• Simplify step-wise
• Systematic approach
• Focus on one operation at a time