Problem 10
Question
For the following exercises, find the inverse of the function on the given domain. $$ f(x)=9-x^{2},[0, \infty) $$
Step-by-Step Solution
Verified Answer
The inverse function is \( f^{-1}(x) = \sqrt{9 - x} \) with domain \( ( -\infty, 9 ] \).
1Step 1: Verify the Function's Domain
The given function is \( f(x) = 9 - x^2 \) with the domain \([0, \infty)\). This means \( x \) can take any value from 0 to infinity.
2Step 2: Express y in Terms of x
To find the inverse, start by expressing the function in terms of \( y \): \[ y = 9 - x^2 \] Our goal is to express \( x \) in terms of \( y \).
3Step 3: Solve for x
Rearrange the equation to solve for \( x \):1. Subtract 9 from both sides: \[ y - 9 = -x^2 \]2. Multiply both sides by -1: \[ 9 - y = x^2 \]3. Take the square root of both sides. Since the domain is \([0, \infty)\) and \( x \) is non-negative: \[ x = \sqrt{9 - y} \]
4Step 4: Write the Inverse Function
The expression \( x = \sqrt{9 - y} \) can be rewritten as a function:\[ f^{-1}(y) = \sqrt{9 - y} \]Replace \( y \) back with \( x \) for the inverse function:\[ f^{-1}(x) = \sqrt{9 - x} \]
5Step 5: Specify the Inverse Function Domain
Since the output of \( f(x) \) is \( y = 9 - x^2 \), and \( x \) is within \([0, \infty)\), \( y \) ranges from 9 down to \( -\infty \). Therefore, the domain of \( f^{-1}(x) \) is \(( -\infty, 9 ]\) to ensure the square root is defined.
Key Concepts
Domain and RangeQuadratic FunctionsSquare Root
Domain and Range
When dealing with functions, understanding the domain and range is essential. The **domain** of a function consists of all the possible input values (typically denoted as \( x \)) for which the function is defined. For the function \( f(x) = 9 - x^2 \), the domain is specified as \([0, \infty)\). This means that you can plug any non-negative number into this function without worrying about undefined expressions.
On the other hand, the **range** is the set of all possible output values the function can produce. For the original function \( f(x) = 9 - x^2 \), as \( x \) starts at 0 and moves towards infinity, the function's values will decrease from 9 to \(-\infty\). Therefore, the range of this function is \(( -\infty, 9 ]\).
When finding the inverse of a function, the roles of the domain and range switch. Thus, the range of the inverse function is \([0, \infty)\), and the domain becomes \(( -\infty, 9 ]\). Understanding these concepts allows for determining what values are possible for inputs and outputs of functions and their inverses.
On the other hand, the **range** is the set of all possible output values the function can produce. For the original function \( f(x) = 9 - x^2 \), as \( x \) starts at 0 and moves towards infinity, the function's values will decrease from 9 to \(-\infty\). Therefore, the range of this function is \(( -\infty, 9 ]\).
When finding the inverse of a function, the roles of the domain and range switch. Thus, the range of the inverse function is \([0, \infty)\), and the domain becomes \(( -\infty, 9 ]\). Understanding these concepts allows for determining what values are possible for inputs and outputs of functions and their inverses.
Quadratic Functions
Quadratic functions, like \( f(x) = 9 - x^2 \), have a characteristic shape known as a parabola when graphed. For the general form \( ax^2 + bx + c \), the coefficient of the \( x^2 \) term dictates the direction the parabola opens. If \( a \) is negative (as in our function where \( a = -1 \)), the parabola opens downward.
This downward-opening shape helps us determine the maximum value of the function. For \( f(x) = 9 - x^2 \), the maximum value is 9, which occurs at \( x = 0 \). As \( x \) increases from 0, the value of \( f(x) \) decreases, forming the downward curve.
Quadratic functions are essential in finding inverses as they often require us to deal with square terms. To invert a quadratic function, you must first express \( x \) in terms of \( y \) and then apply a square root, as illustrated in the step-by-step solution.
This downward-opening shape helps us determine the maximum value of the function. For \( f(x) = 9 - x^2 \), the maximum value is 9, which occurs at \( x = 0 \). As \( x \) increases from 0, the value of \( f(x) \) decreases, forming the downward curve.
Quadratic functions are essential in finding inverses as they often require us to deal with square terms. To invert a quadratic function, you must first express \( x \) in terms of \( y \) and then apply a square root, as illustrated in the step-by-step solution.
Square Root
The square root operation is crucial when finding the inverse of a quadratic function. Given an equation like \( x^2 = a \), the corresponding square roots are \( x = \pm \sqrt{a} \). However, when determining the inverse function or finding specific solutions, focus on the positive square root due to domain considerations.
In the inverse function \( f^{-1}(x) = \sqrt{9 - x} \), the square root of \( 9 - x \) is taken. Since our function's original domain ensures that \( x \) is non-negative, the square root must also be non-negative to keep the function valid over its domain.
The square root function is not defined for negative inputs if we're only considering real numbers, which is why the domain of \( f^{-1}(x) \) was adjusted to \(( -\infty, 9 ]\). This ensures that within the domain, we avoid undefined square roots, maintaining the function's correctness and applicability. Understanding how to manipulate and interpret square roots is essential in solving and inverting quadratic functions.
In the inverse function \( f^{-1}(x) = \sqrt{9 - x} \), the square root of \( 9 - x \) is taken. Since our function's original domain ensures that \( x \) is non-negative, the square root must also be non-negative to keep the function valid over its domain.
The square root function is not defined for negative inputs if we're only considering real numbers, which is why the domain of \( f^{-1}(x) \) was adjusted to \(( -\infty, 9 ]\). This ensures that within the domain, we avoid undefined square roots, maintaining the function's correctness and applicability. Understanding how to manipulate and interpret square roots is essential in solving and inverting quadratic functions.
Other exercises in this chapter
Problem 10
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