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Magazine Chapter 9: Problem 63
Graph function and its inverse using the same set of axes. \(f(x)=x^{3}+1\)
Short Answer
Expert verified
The graphs of \( f(x) = x^3 + 1 \) and its inverse \( f^{-1}(x) = \sqrt[3]{x - 1} \) are reflections over the line \( y = x \).
Step by step solution
01
- Understand the Function
The given function is: \[ f(x) = x^3 + 1 \] This is a cubic function shifted up by 1 unit.
02
- Find the Inverse Function
To find the inverse function, first, replace \( f(x) \) with \( y \): \[ y = x^3 + 1 \]Next, solve for \( x \):Subtract 1 from both sides: \[ y - 1 = x^3 \]Take the cube root of both sides: \[ x = \sqrt[3]{y - 1} \]Now, replace \( y \) with \( f^{-1}(x) \) to get the inverse function: \[ f^{-1}(x) = \sqrt[3]{x - 1} \]
03
- Plot the Original Function
Create a table of values for \( f(x) = x^3 + 1 \) and plot these points on the graph. For example:- When \( x = -2 \), \( f(x) = (-2)^3 + 1 = -7 \)- When \( x = 0 \), \( f(x) = 0^3 + 1 = 1 \)- When \( x = 2 \), \( f(x) = 2^3 + 1 = 9 \)Connect these points to form the graph of \( f(x) \).
04
- Plot the Inverse Function
Create a table of values for \( f^{-1}(x) = \sqrt[3]{x - 1} \) and plot these points on the same graph. For example:- When \( x = -7 \), \( f^{-1}(x) = \sqrt[3]{-7 - 1} = -2 \)- When \( x = 1 \), \( f^{-1}(x) = \sqrt[3]{1 - 1} = 0 \)- When \( x = 9 \), \( f^{-1}(x) = \sqrt[3]{9 - 1} = 2 \)Connect these points to form the graph of \( f^{-1}(x) \).
05
- Verify the Graphs
The graph of \( f(x) \) should be a cubic curve shifted up by 1 unit, and the graph of \( f^{-1}(x) \) should be its reflection over the line \( y = x \). Ensure both graphs intersect along the line \( y = x \) to verify correctness.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cubic Function
A cubic function is a type of polynomial function where the highest degree term is raised to the power of three. In mathematical terms, it is generally written as: \[ f(x) = ax^3 + bx^2 + cx + d \]where 'a', 'b', 'c', and 'd' are constants, and the term with the highest degree is 3 (i.e., the cubic term). An example of a cubic function in this context is \[ f(x) = x^3 + 1 \]. This specific function represents a standard cubic curve that has been shifted upwards by 1 unit on the y-axis.
Some important characteristics of cubic functions include:
Some important characteristics of cubic functions include:
- They can have one, two or three real roots (the x-values where the function intersects the x-axis).
- Their graphs can show turning points (local maxima and minima) and a general 'S' shape.
- As x approaches positive or negative infinity, the function value also gets progressively larger or smaller respectively.
Graphing Functions
Graphing functions involves plotting various points on a set of axes to visually represent the function's behavior. Here's how you can graph the function \[ f(x) = x^3 + 1 \]:
- First, select a range of x-values (e.g., -2, -1, 0, 1, 2) to substitute into the function.
- Calculate the corresponding y-values using these x-values. For instance, if x = -2, then y = (-2)^3 + 1 = -7.
- Plot these (x, y) points on the grid.
- For more complex functions, it's helpful to identify key features like intercepts (where the function crosses the axes) and turning points to guide the graphing process.
- Consider the function's end behavior as x approaches positive and negative infinity to ensure the graph's tails reflect its true behavior.
Finding the Inverse
Finding the inverse of a function involves reversing the original function's operations. For the cubic function \[ f(x) = x^3 + 1 \], the inverse is found as follows:
This inverse function \[ f^{-1}(x) \] essentially 'undoes' the operations performed by the original function \[ f(x) \]. To graph the inverse on the same axes:
- Start by replacing \[ f(x) \] with \[ y \]: \[ y = x^3 + 1 \]
- Solve for \[ x \]: Subtract 1 from both sides to isolate the cubic term: \[ y - 1 = x^3 \]
- Take the cube root of both sides: \[ x = \sqrt[3]{y - 1} \]
- Finally, replace \[ y \] with \[ x \] and \[ x \] with \[ f^{-1}(x) \] to express the inverse: \[ f^{-1}(x) = \sqrt[3]{x - 1} \]
This inverse function \[ f^{-1}(x) \] essentially 'undoes' the operations performed by the original function \[ f(x) \]. To graph the inverse on the same axes:
- Select various x-values for \[ f^{-1}(x) \] and compute their corresponding y-values.
- Plot these (x, y) points.
