/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 A function \(f\) is given. (a) S... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 63

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=\sqrt{x+1} $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = x^2 - 1 \), where \( x \geq 0 \).

Step by step solution

01

Understanding the Function

The function given is \( f(x) = \sqrt{x+1} \). This is a square root function that shifts the basic square root graph \( \sqrt{x} \) to the left by 1 unit. The domain of the function is \( x \geq -1 \) and the range is \( y \geq 0 \).
02

Sketching the Graph of f

To sketch \( f(x) = \sqrt{x+1} \), start by considering a few key points: For \( x = -1, f(x) = 0 \); for \( x = 0, f(x) = 1 \); and for \( x = 3, f(x) = 2 \). Plot these points and draw a smooth curve that starts at \( (-1, 0) \) and moves rightward, becoming flatter as it extends.
03

Understanding the Inverse Function

The inverse function, \( f^{-1}(y) \), is obtained by swapping the roles of \( x \) and \( y \) in the original function equation, reflecting across the line \( y = x \). The domain of \( f^{-1} \) is the range of \( f \), \( y \geq 0 \); and the range of \( f^{-1} \) is the domain of \( f \), \( x \geq -1 \).
04

Sketching the Graph of f^{-1}

To sketch the graph of \( f^{-1} \), reflect the graph of \( f(x) = \sqrt{x+1} \) across the line \( y = x \). This involves plotting points where coordinates are swapped - like \( (0, -1) \), \( (1, 0) \), and \( (2, 3) \) - and joining them smoothly just like the curve of \( f \).
05

Finding the Algebraic Expression for f^{-1}

Start with the equation \( y = \sqrt{x+1} \) and solve for \( x \):1. Square both sides: \( y^2 = x + 1 \).2. Solve for \( x \): \( x = y^2 - 1 \).3. Replace \( x \) with \( f^{-1}(y) \) and \( y \) with \( x \):\( f^{-1}(x) = x^2 - 1 \).This represents the inverse function, with domain \( x \geq 0 \).

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Key Concepts

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

Square Root Functions
Square root functions, like the one given in the problem, are represented by equations in the form of \( f(x) = \sqrt{x} \). In our problem, the function is \( f(x) = \sqrt{x+1} \). This function shifts the basic square root graph \( \sqrt{x} \) horizontally. Specifically, it moves to the left by 1 unit.

Let's break it down a bit more:
  • The square root function starts at a certain point on the x-axis and increases slowly.
  • For \( x \geq -1 \), \( f(x) \) produces real and positive values, making it possible to plot these points on a graph.
  • The function curve becomes flatter as \( x \) increases. This means it doesn't rise as steeply after the first few values.
When sketching this graph, you start at the point \( (-1, 0) \) and move rightwards, forming a gentle upward curve. This visual representation helps in understanding how the square root function behaves over its domain.
Graph Reflection
Graph reflection is a crucial concept when it comes to inverse functions. In the specific case of the function \( f(x) = \sqrt{x+1} \), we need to understand how to reflect it to find its inverse \( f^{-1}(x) \). This involves a few key steps:

  • First, swap the roles of \( x \) and \( y \). For example, if you have a point \( (a, b) \) on the graph of \( f(x) \), then the inverse reflects it to \( (b, a) \).
  • Reflect the entire graph across the line \( y = x \). This is a diagonal line where each point mirrors over.
For the problem at hand, Swap some coordinates from the graph of \( f(x) = \sqrt{x+1} \), like \( (-1, 0), (0, 1) \), resulting in inverse graph points \( (0, -1), (1, 0) \).
It's important to note that reflecting across \( y = x \) means you're creating a mirror image of the original graph, allowing us to visualize the inverse function.
Domain and Range
Domain and range are fundamental concepts for understanding any function and its inverse. Let's quickly differentiate these concepts for both original and inverse functions:

  • **Domain**: This is the set of all possible input values (\( x \)) for which the function is defined.
  • **Range**: This is the set of all possible output values (\( y \)) that the function can produce.
For our function \( f(x) = \sqrt{x+1} \):

  • The domain is \( x \geq -1 \) because values less than \(-1\) would result in a negative inside the square root, which is undefined in the set of real numbers.
  • The range is \( y \geq 0 \) because square roots produce non-negative values.
Now, if we look at the inverse \( f^{-1}(x) = x^2 - 1 \):

  • The domain of \( f^{-1} \) (originating from the range of \( f \)) is \( x \geq 0 \).
  • The range of \( f^{-1} \) (originating from the domain of \( f \)) is \( y \geq -1 \).
Understanding these allows you to correctly identify and describe the nature of both the function and its inverse.

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Most popular questions from this chapter

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ f(x)=2+x $$

Multiple Discounts \(A\) car dealership advertises a 15\(\%\) discount on all its new cars. In addition, the manufacturer offers a \(\$ 1000\) rebate on the purchase of a new car. Let \(x\) represent the sticker price of the car. (a) Suppose only the 15\(\%\) discount applies. Find a function \(f\) that models the purchase price of the car as a function of the sticker price \(x .\) (b) Suppose only the \(\$ 1000\) rebate applies. Find a function \(g\) that models the purchase price of the car as a function of the sticker price \(x\) (c) Find a formula for \(H=f \circ g .\) (d) Find \(H^{-1} .\) What does \(H^{-1}\) represent? (e) Find \(H^{-1}(13,000) .\) What does your answer represent?

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=x-3, \quad g(x)=x^{2} $$

The given function is not one-to-one. Restrict its domain so that the resulting function \(i s\) one-to-one. Find the inverse of the function with the restricted domain. (There is more than one correct answer.) $$ f(x)=4-x^{2} $$

Revenue, cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per bumper sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that revenue \(=\) price per tiem \(\times\) number of items sold to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x\)
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