Problem 46
Question
Find the inverse function of \(f\). \(f(x)=\left(2-x^{3}\right)^{5}\)
Step-by-Step Solution
Verified Answer
The inverse function is \( f^{-1}(x) = (2 - x^{1/5})^{1/3} \).
1Step 1: Understanding the Problem
We are given the function \( f(x) = (2-x^3)^5 \) and need to find the inverse function, \( f^{-1}(x) \). The inverse function should satisfy \( f(f^{-1}(x)) = x \).
2Step 2: Set Up the Equation for Inversion
To find the inverse, start by writing \( y = (2-x^3)^5 \). We want to express \( x \) in terms of \( y \).
3Step 3: Apply the Fifth Root
To isolate \( x \), take the fifth root of both sides: \[ y^{1/5} = 2 - x^3 \].
4Step 4: Isolate the Cubed Term
Subtract 2 from both sides to isolate \( x^3 \): \[ x^3 = 2 - y^{1/5} \].
5Step 5: Solve for x
Solve for \( x \) by taking the cube root of both sides: \[ x = (2 - y^{1/5})^{1/3} \].
6Step 6: Rewrite the Inverse Function
The inverse function \( f^{-1}(x) \) is finally given by: \[ f^{-1}(x) = (2 - x^{1/5})^{1/3} \].
Key Concepts
Cube RootFifth RootAlgebraMathematical Functions
Cube Root
The concept of a cube root is fundamental in mathematics, especially when dealing with powers and inverse operations. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
Cube roots are important in simplifying equations and expressions, particularly in algebra and calculus. They are crucial in finding inverse functions, like in our original exercise, where solving for \( x \) required taking the cube root.
Knowing cube roots enables:
Cube roots are important in simplifying equations and expressions, particularly in algebra and calculus. They are crucial in finding inverse functions, like in our original exercise, where solving for \( x \) required taking the cube root.
Knowing cube roots enables:
- Solving polynomial equations of higher degrees
- Simplifying complex algebraic expressions
- Understanding geometric concepts, like the volume of cubes
Fifth Root
The fifth root is another critical concept that helps simplify expressions involving larger exponents. It is the value that, when raised to the fifth power, results in the original number. For example, the fifth root of 32 is 2, because \( 2^5 = 32 \).
Finding fifth roots is essential in many mathematical problems, particularly in equations where powers of five are involved. In our exercise, discovering the inverse function demanded taking the fifth root in order to simplify the equation \( y^{1/5} = 2 - x^3 \). Using fifth roots can:
Finding fifth roots is essential in many mathematical problems, particularly in equations where powers of five are involved. In our exercise, discovering the inverse function demanded taking the fifth root in order to simplify the equation \( y^{1/5} = 2 - x^3 \). Using fifth roots can:
- Help solve equations with higher-degree polynomials
- Facilitate finding solutions to algebraic problems
- Simplify expressions in calculus and physics tasks
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is essential for understanding and solving equations and expressions. In algebra, you'll often work with unknown quantities represented by variables like \( x \) or \( y \).
Our original problem required knowledge of algebra to solve for the inverse function step-by-step. Inverse functions involve flipping \( x \) and \( y \) to find a relationship that results in an expression for \( x \) given \( y \). In algebra, critical skills include:
Our original problem required knowledge of algebra to solve for the inverse function step-by-step. Inverse functions involve flipping \( x \) and \( y \) to find a relationship that results in an expression for \( x \) given \( y \). In algebra, critical skills include:
- Manipulating equations to isolate variables
- Understanding the properties of equality
- Solving various types of equations (linear, quadratic, polynomial)
Mathematical Functions
Mathematical functions are fundamental in expressing relationships between variables. A function associates each input with exactly one output. For instance, in the function \( f(x) = (2-x^3)^5 \), every \( x \) value corresponds to one unique \( y \) value.
The study of functions covers understanding their properties, such as domain and range, which are essential in calculus and various applications. The inverse function, \( f^{-1}(x) \), is a vital concept where the roles of inputs and outputs of the original function are reversed.
Key features of mathematical functions include:
The study of functions covers understanding their properties, such as domain and range, which are essential in calculus and various applications. The inverse function, \( f^{-1}(x) \), is a vital concept where the roles of inputs and outputs of the original function are reversed.
Key features of mathematical functions include:
- Understanding different types of functions (linear, polynomial, exponential)
- Graphing functions to visualize relationships
- Utilizing functions to model real-world scenarios
Other exercises in this chapter
Problem 46
Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{1-x^{2}} & {\text { if } x \leq 2} \\ {x} & {\text { if } x>2}\end{array}\
View solution Problem 46
33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x-1|
View solution Problem 46
Find the domain of the function. $$ f(x)=\sqrt[4]{x+9} $$
View solution Problem 47
\(45-50\) Express the function in the form \(f \circ g\) $$ G(x)=\frac{x^{2}}{x^{2}+4} $$
View solution