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Chapter 5: Problem 10

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=9-x^{2},[0, \infty) $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(y) = \sqrt{9 - y} \) with domain \( [0, 9] \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = 9 - x^2 \) with the domain \([0, \infty)\). This means \( x \) can take any non-negative real number. This ensures that the function is not multi-valued over this domain, as it's a part of a downward-opening parabola restricted to its non-negative half.
02

Set Up the Equation for Inversion

To find the inverse, swap \( f(x) \) and \( x \). In other words, let \( y = 9 - x^2 \). For inversion, solve for \( x \) in terms of \( y \): \( x = 9 - y \).
03

Solve for x in Terms of y

Rearrange the equation \( y = 9 - x^2 \) to solve it for \( x \). We have: \( x^2 = 9 - y \). Thus, \( x = \sqrt{9 - y} \). Since the domain of \( x \) is non-negative, take the positive root.
04

Define the Inverse Function

The inverse function is where you solved for \( x \) in terms of \( y \). Therefore, the inverse function is \( f^{-1}(y) = \sqrt{9 - y} \). Note that the range of \( f^{-1}(y) \) is \([0, 3]\) because \( y \) in the function \( f(x) = 9 - x^2 \) varies from 9 to 0 as \( x \) goes from 0 to 3.

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Key Concepts

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

Domain and Range
In the realm of functions, the terms **domain** and **range** have specific meanings that help us understand what values the function can accept and produce. Let's say we're working with a function like \[ f(x) = 9 - x^2 \]
  • Domain: This refers to all the possible values that can be plugged into the function for the variable, usually referred to as \( x \). For the above function, the domain is given as \([0, \infty)\). This means \( x \) can take all non-negative real numbers. The domain is crucial because it ensures the function is not ambiguous, or multi-valued, even though squared terms generally produce parabola shapes that can be symmetrical.
  • Range: These are the possible values that the function can output, known as \( f(x) \). For \[ f(x) = 9 - x^2 \] with the specified domain, the range becomes \([0, 9]\). As \( x \) increases from 0 to a higher value, \( f(x) \) decreases from 9 to 0, defining its specific range.
Understanding the domain and range helps us grasp the behavior of functions more thoroughly.
Function Inversion
Function inversion is taking a function and flipping it over its domain and range, producing its 'opposite' or inverse function. If we start with a function like \[ f(x) = 9 - x^2 \] function inversion involves switching the input \( x \) with the output \( f(x) = y \).Here's how we find the inverse:
  • Step 1: Replace \( f(x) \) with \( y \), giving us \[ y = 9 - x^2 \].
  • Step 2: Solve for \( x \) in terms of \( y \): Rearrange the equation to \[ x^2 = 9 - y \], then solve for \( x \) to get \[ x = \sqrt{9 - y} \].
  • Step 3: The positive square root is taken because the domain \([0, \infty)\) ensures \( x \) values are non-negative.
  • Step 4: The inverse function is then \[ f^{-1}(y) = \sqrt{9 - y} \].
What this inversion does is give us a new function where \( y \) as input returns what \( x \) was in the original function.
Quadratic Functions
Quadratic functions are among the most fundamental types of polynomial functions. They have a standard form that looks like this: \[ f(x) = ax^2 + bx + c \].These functions graph as parabola shapes, which can either open upwards or downwards, depending on the sign of the \( a \) coefficient.For the quadratic function \[ f(x) = 9 - x^2 \], we observe the following:
  • Downward Opening: The negative sign before \( x^2 \) indicates that the parabola opens downward.
  • Vertex Formed: This specific type, \[ f(x) = 9 - x^2 \], can also be rewritten as \[ x^2 - 9 \]. This reveals its vertex, located at (0,9), which is its maximum point due to the downward opening.
  • Symmetry: Quadratics are symmetric around their vertex. In our case, the function's restriction to \([0, \infty)\) removes the left half of the symmetry, simplifying analysis and ensuring non-negative value handling.
Understanding quadratic functions allows us to predict graph behavior, solve equations graphically, and appreciate varied algebraic structures.

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Most popular questions from this chapter

For the following exercises, use a graph to help determine the domain of the functions. $$ f(x)=\sqrt{\frac{x(x+3)}{x-4}} $$

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as the square root of \(w\) and the square of \(t\). When \(x=3, z=1, w=25,\) and \(t=2\) then \(y=6\).

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)

For the following exercises, use the given information to answer the questions. The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth's center, what would it weigh it were 3970 miles from Earth's center?

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as \(x\) and whe \(x=4, \quad y=2\).
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