Question

Find the inverse of each of the following functions: (a) \(f(x)=4 x+7\) (b) \(f(x)=x\) (c) \(f(x)=-23 x\) (d) \(f(x)=\frac{1}{x+1}\)

Short answer

Question: Find the inverse of each function. a) f(x) = 4x + 7 b) f(x) = x c) f(x) = -23x d) f(x) = \(\frac{1}{x+1}\) Answer: a) \(f^{-1}(x)=\frac{x-7}{4}\) b) \(f^{-1}(x)=x\) c) \(f^{-1}(x)=-\frac{x}{23}\) d) \(f^{-1}(x)=\frac{-x+1}{x-1}\)

Step-by-step solution

(a) Find the inverse of the function f(x) = 4x + 7.

To find the inverse of the function, swap x with y and solve for y: 1. Replace f(x) with y: \(y=4x+7\) 2. Swap x and y: \(x=4y+7\) 3. Solve for y: \(y=\frac{x-7}{4}\) The inverse of the function f(x) = 4x + 7 is \(f^{-1}(x)=\frac{x-7}{4}\)

(b) Find the inverse of the function f(x) = x.

To find the inverse of the function, swap x with y and solve for y: 1. Replace f(x) with y: \(y=x\) 2. Swap x and y: \(x=y\) The inverse of the function f(x) = x is \(f^{-1}(x)=x\)

(c) Find the inverse of the function f(x) = -23x.

To find the inverse of the function, swap x with y and solve for y: 1. Replace f(x) with y: \(y=-23x\) 2. Swap x and y: \(x=-23y\) 3. Solve for y: \(y=-\frac{x}{23}\) The inverse of the function f(x) = -23x is \(f^{-1}(x)=-\frac{x}{23}\)

(d) Find the inverse of the function f(x) = \(\frac{1}{x+1}\).

To find the inverse of the function, swap x with y and solve for y: 1. Replace f(x) with y: \(y=\frac{1}{x+1}\) 2. Swap x and y: \(x=\frac{1}{y+1}\) 3. Solve for y: Firstly, we can remove the fraction by multiplying both sides by \((y+1)\): \(x(y+1)=1\) 4. Now, expand the equation: \(xy+x=1\) 5. To isolate y, move \((x-1)\) to the other side and factor out y: \(y(x-1)=-x+1\) 6. Lastly, divide by \((x-1)\) to solve for y: \(y=\frac{-x+1}{x-1}\) The inverse of the function f(x) = \(\frac{1}{x+1}\) is \(f^{-1}(x)=\frac{-x+1}{x-1}\)

Key concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental technique used to rearrange and simplify mathematical expressions and equations. It involves the application of various rules and properties of algebra, such as the distributive property, the commutative and associative properties, and the rules for dealing with exponents.

For instance, when finding the inverse of a function like

• Original function:
  Mathematically, it's represented as
  \( f(x) = 4x + 7 \).
  
• Re-expressing the function:
  We first write it in terms of another variable, say 'y', which gives us
  \( y = 4x + 7 \).
  
• We then swap 'x' and 'y' to make 'x' the subject, leading to
  \( x = 4y + 7 \).
  
• Finally, we solve for 'y' which involves isolating 'y' on one side of the equation. To do this, we subtract '7' from both sides and then divide by '4', resulting in
  \( y = \frac{x - 7}{4} \).
  

Each transformation represents an algebraic manipulation aimed at simplifying the expression and ultimately finding the inverse function.

Function Inversion

Function inversion is a process by which we determine the inverse of a given function. The inverse of a function
\( f \)
is denoted as
\( f^{-1} \)
and it essentially reverses the original function such that if
\( f(a) = b \), then
\( f^{-1}(b) = a \).

To find the inverse, we perform a 'role-reversal' where the input becomes the output and vice versa. This means we replace
\( f(x) \)
with
\( y \)
and then swap
\( x \)
and
\( y \)
in the equation. By solving this new equation for
\( y \),
we find the inverse function. This procedure is crucial as it reveals the behavior of a function in reverse, allowing us to understand the relationship between inputs and outputs from both perspectives.

Solving Equations

Solving equations is a core concept within algebra which involves finding the value(s) of variables that satisfy the given equation. We use a variety of techniques such as isolating the variable, factoring, using inverses, and applying operations equally to both sides of an equation, to maintain the balance.

Consider the equation that comes from function inversion where we have
\( xy + x = 1 \).
To solve for
\( y \),
we perform several steps:

Isolation of Variable

• First, we rearrange the terms to group all terms containing the variable
  \( y \)
  on one side, leading to
  \( y(x-1) = -x+1 \).
• Next, we factor out the variable
  \( y \)
  on the left-hand side to better isolate it.
• Finally, we divide through by
  \( (x-1) \)
  to complete the isolation, giving us the solved variable
  \( y = \frac{-x+1}{x-1} \).

It is a systematic approach that allows us to find unknown variables and is particularly useful when checking solutions for functions and their inverses.