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Chapter 10: Problem 44

a) Determine whether it is one-to-one. [5.1] b) If it is one-to-one, find a formula for the inverse. [ 5.1] $$f(x)=\sqrt{x+4}$$

Short Answer

Expert verified
The function is one-to-one. The inverse is \( f^{-1}(x) = x^2 - 4 \).

Step by step solution

01

Determine if the function is one-to-one

Check whether each output is produced by exactly one input. For the function \( f(x) = \sqrt{x+4} \), consider two values of \( x \), say \( x_1 \) and \( x_2 \). If \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), then the function is one-to-one. \( f(x_1) = \sqrt{x_1 + 4} \) and \( f(x_2) = \sqrt{x_2 + 4} \).If \( \sqrt{x_1 + 4} = \sqrt{x_2 + 4} \), then \( x_1 + 4 = x_2 + 4 \) so \( x_1 = x_2 \). Hence, the function is one-to-one.
02

Express the function in terms of y

If the function is one-to-one, find the inverse by first expressing the function in terms of \( y \).Set \( y = f(x) \).So, we have: \[ y = \sqrt{x + 4} \]
03

Solve for x

Square both sides to remove the square root. \[ y^2 = x + 4 \]
04

Isolate x

Subtract 4 from both sides to solve for \( x \): \[ x = y^2 - 4 \]
05

Write the inverse function

Switch \( x \) and \( y \) to write the inverse function: \[ f^{-1}(x) = x^2 - 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Understanding one-to-one functions is crucial in precalculus and calculus. These functions have a unique output for every unique input. To determine if a function is one-to-one, you can use the horizontal line test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one. For example, with the function \( f(x) = \sqrt{x+4} \), consider two values \( x_1 \) and \( x_2 \). If \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), the function is one-to-one. Here, \( \sqrt{x_1 + 4} = \sqrt{x_2 + 4} \) leads to \( x_1 + 4 = x_2 + 4 \) and hence, \( x_1 = x_2 \). Therefore, \( f(x) = \sqrt{x+4} \) is one-to-one.
headline of the respective core concept
Precalculus is the foundation for understanding more advanced mathematics. One of its core concepts is the study of functions and their properties. When dealing with functions like \( f(x) = \sqrt{x+4} \), it's important to understand the domain and range, which help determine where the function is defined and its possible outputs. In this function, the domain is \( x \ge -4 \) because the expression inside the square root must be non-negative. The range is \( y \ge 0 \) because square roots always produce non-negative results. Visualizing these in graphs can help you grasp how the function behaves prior to solving for its inverse.
headline of the respective core concept
Solving equations often involves isolating the variable of interest. To find an inverse function, you start by expressing the function with a variable, then solve for the other variable. For our function \( f(x) = \sqrt{x+4} \), we set it as \( y = \sqrt{x+4} \). Next, we solve for \( x \) by first squaring both sides to get rid of the square root: \[ y^2 = x + 4 \]. Then, we isolate \( x \) by subtracting 4 from both sides: \[ x = y^2 - 4 \]. These steps are common in many functions you'll encounter in math, and mastering them lets you tackle many variations of solving for inverse functions.
headline of the respective core concept
An inverse function essentially 'reverses' the effect of the original function. When we have a one-to-one function like \( f(x) = \sqrt{x+4} \), finding its inverse involves solving the equation for \( x \) and then swapping \( x \) and \( y \). This process gives \( f^{-1}(x) \). From our steps, we derived \( x = y^2 - 4 \) and by switching \( x \) and \( y \), we get the inverse: \[ f^{-1}(x) = x^2 - 4 \]. This inverse function lets you find the original input \( x \) when given the output. Understanding and deriving inverse functions is a vital skill in algebra and precalculus, laying the groundwork for understanding more complex functional relationships in calculus.

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