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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.1.46
Chapter 1, Problem 1.1.46

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = (−x)²/³

Verified step by step guidance
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First, understand the function y = (-x)^(2/3). This function involves a fractional exponent, which can be rewritten as y = ((-x)^2)^(1/3). This means we are taking the cube root of the square of -x.
Next, consider the symmetry of the function. A function is symmetric about the y-axis if f(x) = f(-x) for all x in the domain. Substitute -x into the function: y = ((-(-x))^2)^(1/3) = (x^2)^(1/3). Since this is equal to the original function, the graph is symmetric about the y-axis.
To determine where the function is increasing or decreasing, find the derivative of y with respect to x. Use the chain rule: if y = u^(1/3) where u = (-x)^2, then dy/dx = (1/3)u^(-2/3) * du/dx. Calculate du/dx = 2(-x)(-1) = 2x.
Substitute du/dx into the derivative: dy/dx = (1/3)((-x)^2)^(-2/3) * 2x. Simplify this expression to find the critical points where the derivative is zero or undefined, which will help identify intervals of increase and decrease.
Analyze the sign of the derivative on intervals determined by the critical points. If dy/dx > 0, the function is increasing on that interval; if dy/dx < 0, the function is decreasing. Use this information to specify the intervals of increase and decrease for the function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing if the value at the second point is less than at the first. Identifying these intervals involves analyzing the first derivative of the function, where positive values indicate increasing behavior and negative values indicate decreasing behavior.
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Determining Where a Function is Increasing & Decreasing

Graphing Functions

Graphing a function involves plotting its points on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, turning points, and asymptotes. For the function y = (−x)²/³, understanding its shape and symmetry is crucial, as it helps in determining where the function increases or decreases and how it behaves at different values of x.
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Graph of Sine and Cosine Function

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits a mirror-like behavior about a specific axis or point. For example, a function is even if f(x) = f(-x) for all x, indicating symmetry about the y-axis, and odd if f(-x) = -f(x), indicating symmetry about the origin. Analyzing the symmetry of the function y = (−x)²/³ can provide insights into its overall shape and the intervals of increase and decrease.
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Properties of Functions
Related Practice
Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the function f (x) = sin 2x + cos 3x.

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Textbook Question

Algebraic Combinations


In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.


f(x) = √(x + 1), g(x) = √(x − 1)

Textbook Question

[Technology Exercise]


a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which


x/2 > 1 + 4/x


b. Confirm your findings in part (a) algebraically.

Textbook Question

Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


y = x + (1/10) sin 30x

Textbook Question

Shifting Graphs


The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.


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Textbook Question

Shifting Graphs


Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.


y = x³ Left 1, down 1