Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = (x - 1)^2 - 1 \).

Step 1: Understand the Problem

The function given is \( f(x) = 1 + \sqrt{1+x} \). To find its inverse, we need to solve for \( x \) in terms of \( y \), where \( y = f(x) \). The goal is to express \( x \) as a function of \( y \).

Step 2: Set Up the Equation

First, rewrite the function in the form \( y = 1 + \sqrt{1+x} \). We will solve this equation for \( x \).

Step 3: Isolate the Square Root

Subtract 1 from both sides to isolate the square root: \( y - 1 = \sqrt{1+x} \).

Step 4: Remove the Square Root

Square both sides of the equation to eliminate the square root: \( (y - 1)^2 = 1 + x \).

Step 5: Solve for x

Subtract 1 from both sides to solve for \( x \): \( x = (y - 1)^2 - 1 \).

Step 6: Write the Inverse Function

The inverse function \( f^{-1}(y) \) is given by the expression we found for \( x \): \( f^{-1}(y) = (y - 1)^2 - 1 \). Since \( f^{-1}(y) = x \), we replace \( y \) with \( x \) in the final answer, giving \( f^{-1}(x) = (x - 1)^2 - 1 \).

Key Concepts

Square Root Functions

Square root functions are essential mathematical expressions that involve the square root symbol, \( \sqrt{} \), which denotes the principal square root of a number. The general form is \( f(x) = \sqrt{x} \), but in our specific exercise, the function is \( f(x) = 1 + \sqrt{1+x} \). This means we are adding 1 to the square root of the expression \( 1+x \).

These functions often represent the inverse of quadratic functions, and they play a crucial role in various fields, such as physics and engineering. Square root functions have a domain restriction because we can't take the square root of a negative number without involving complex numbers.

For our function \( f(x) = 1 + \sqrt{1+x} \):

• The domain, where the function is defined, is \( x \geq -1 \) because \( 1+x \) must be non-negative.
• The range is all real numbers greater than or equal to 1, because \( 1 + \sqrt{1+x} \) starts at 1 and increases without bound as \( x \) increases.

This understanding helps when you look for an inverse function.

Solving Equations

Solving equations involves finding the values for variables that satisfy the given mathematical condition. In the case of finding an inverse function, we switch the dependent and independent variables, solve for the original independent variable, and express it in terms of the new dependent variable.

Here, starting with the equation \( y = 1 + \sqrt{1+x} \), our task was to solve for \( x \) in terms of \( y \).

The process of solving this equation involved several steps:

• First, isolate the square root by subtracting 1 from both sides to get \( y - 1 = \sqrt{1+x} \).
• Eliminate the square root by squaring both sides, resulting in \( (y - 1)^2 = 1 + x \).
• Solve for \( x \) by rearranging the equation to \( x = (y - 1)^2 - 1 \).

Completing these steps meticulously ensures that you correctly determine the inverse function, allowing functions to map values backward with precision.

Inverse Operations

Inverse operations are paired mathematical processes that undo each other. Understanding these operations is crucial for finding an inverse function. In the context of our task, the square root and squaring are inverse operations.

When you take a number and apply a square root, and then square the result, you essentially return to your original number—provided all conditions are met for real values. This inverse relationship is key to finding the inverse function of \( f(x) = 1 + \sqrt{1+x} \).

Let's break down how these inverse operations work in our solution:

• Once we isolated the square root with \( y - 1 = \sqrt{1+x} \), squaring both sides counteracts the square root.
• This led to \( (y-1)^2 \), effectively 'removing' the square root and simplifying to \( = 1 + x \).

Through careful application of these inverse operations, solving for \( x \) becomes straightforward, leading us to determine the inverse function: \( f^{-1}(x) = (x - 1)^2 - 1 \). Understanding these operations ensures clarity and precision in mathematical problem-solving.