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Chapter 2: Problem 64

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 \(f^{-1}\). $$f(x)=x^{3}-1$$

Short Answer

Expert verified
Inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step by step solution

01

Understanding the Function

The given function is \( f(x) = x^3 - 1 \). This is a cubic function that has been translated one unit down. The basic cubic function \( x^3 \) is symmetric about the origin, and the translation modifies it.
02

Sketching the Graph of \( f \)

Plot the graph of \( f(x) = x^3 - 1 \). It starts at \( (-1, 0) \) on the y-axis, with the point \( x = 0 \) mapped to \( f(0) = -1 \). As \( x \) increases or decreases, the graph follows the typical cubic shape, flattening around the origin.
03

Understanding the Inverse Function

To find the inverse function, \( f^{-1} \), we need to swap the roles of \( x \) and \( y \). So, for \( f(x) = x^3 - 1 \), the inverse will reverse the operation of subtracting 1 and then taking the cube root.
04

Sketching the Graph of \( f^{-1} \)

To sketch \( f^{-1} \), reflect the graph of \( f \) across the line \( y = x \). The graph of \( f^{-1} \) will pass through the points \( (0, -1) \), \( (-1, 0) \), and it will have an inverse cubic shape.
05

Finding \( f^{-1} \) Algebraically

Starting with \( y = x^3 - 1 \), solve for \( x \) in terms of \( y \). First, add 1 to both sides to obtain \( y + 1 = x^3 \). Then, take the cube root of both sides to find \( x = \sqrt[3]{y + 1} \). Therefore, the inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

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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 polynomial of degree 3 and often takes the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). The simplest form of a cubic function is \( f(x) = x^3 \), which is symmetric about the origin.

Cubic functions display a few distinct characteristics:
  • They can have one or three real roots.
  • Their graphs exhibit points of inflection, where the curve changes concavity.
  • The end behavior means as \( x \to \pm \infty \), \( f(x) \to \pm \infty \) respectively if \( a > 0 \), or \( f(x) \to \mp \infty \) if \( a < 0 \).
In our exercise, the function \( f(x) = x^3 - 1 \) is derived from the basic cubic \( x^3 \) by a vertical shift downward by one unit. This modification affects its position but not its overall shape. The inflection point of this cubic is at \( (0, -1) \), reflecting that vertical shift.
Graph Sketching
Graph sketching is a crucial strategy in understanding the overall behavior of functions. For a cubic function like \( f(x) = x^3 - 1 \), starting with key characteristics helps make the sketch more accurate.

Here's how to approach sketching:
  • Identify Critical Points: Find where the function intersects the axes. For \( f(x) = x^3 - 1 \), the intersection is at points like \( (0, -1) \).
  • Analyze Behavior Near the Origin: Notice how the function behaves around \( x = 0 \), since cubic functions tend to flatten near inflection points.
  • Examine End Behavior: As \( x \) heads towards \( \pm \infty \), \( f(x) \) carries the graph upwards or downwards depending on the lead coefficient.
  • Enhance with Specific Points: Calculating values for specific \( x \) values can provide additional plot points to ensure accuracy.
This approach ensures a robust sketch that reflects the function鈥檚 behavior, which is essential when later reflecting it to sketch its inverse.
Function Transformation
Function transformations allow us to modify the basic graph of functions in predictable ways. The expression \( f(x) = x^3 - 1 \) reflects a transformation of the base function \( x^3 \) by shifting it one unit downwards.

Function transformations primarily include:
  • Vertical Shifts: Adding or subtracting a constant shifts the graph up or down. \( x^3 - 1 \) means shifting downward.
  • Horizontal Shifts: Adding or subtracting inside the function argument would shift the graph left or right, but there is none here.
  • Stretching/Compressing: Multiplying the function or variable by constants influences its steepness and overall spread.
Comprehending these transformations is crucial when altering functions for graphing and when working with inverse operations. For instance, understanding the shift helps when sketching \( f^{-1} \), by ensuring the reflection is accurately mirrored across the line \( y = x \). This graphical operation requires identifying points affected by these transformations.

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Most popular questions from this chapter

The cost \(C\) in dollars of producing \(x\) yards of a certain fabric is given by the function. $$C(x)=1500+3 x+0.02 x^{2}+0.0001 x^{3}$$ (a) Find \(C(10)\) and \(C(100)\) (b) What do your answers in part (a) represent? (c) Find \(C(0)\). (This number represents the fixed costs.)

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\) Subtract \(5,\) then square

Find the domain of the function. $$f(t)=\sqrt[3]{t}-1$$

Find the domain of the function. $$f(x)=\frac{x^{4}}{x^{2}+x-6}$$

When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2} .\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.
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