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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 51
Capitolo 1, Problema 51

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.


ƒ₁(x)        ƒ₂(x)
 __         ___
√ x         √ |x|

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1
Identify the functions: f₁(x) = √x and f₂(x) = √|x|. The function f₁(x) is defined for x ≥ 0, while f₂(x) is defined for all real numbers because the absolute value ensures the input to the square root is non-negative.
Graph f₁(x) = √x: This graph is a half-parabola starting at the origin (0,0) and extending to the right, increasing slowly as x increases.
Graph f₂(x) = √|x|: This graph is symmetric about the y-axis. For x ≥ 0, it matches f₁(x), and for x < 0, it mirrors the right side of the graph across the y-axis, forming a V-shape.
Compare the graphs: The graph of f₂(x) = √|x| includes both the positive and negative x-values, creating a reflection of f₁(x) across the y-axis. This is due to the absolute value, which makes the input to the square root non-negative, allowing the function to be defined for negative x-values as well.
Describe the effect of the absolute value: Applying the absolute value to the input of f₂(x) extends the domain of the function to include negative x-values, resulting in a graph that is symmetric about the y-axis, unlike f₁(x) which is only defined for non-negative x-values.

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Absolute Value Function

The absolute value function, denoted as |x|, transforms any negative input into its positive counterpart while leaving positive inputs unchanged. This means that for any x, |x| = x if x ≥ 0 and |x| = -x if x < 0. When applied to a function, it reflects any portion of the graph that lies below the x-axis to above the x-axis, effectively altering the function's behavior in the negative domain.
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Average Value of a Function

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values). The shape and position of the graph provide insights into the function's behavior, such as its intercepts, increasing or decreasing intervals, and overall trends. Understanding how to graph functions is essential for comparing different functions, like ƒ₁ and ƒ₂ in this case.
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Graph of Sine and Cosine Function

Transformation of Functions

Transformations of functions refer to changes made to the original function that affect its graph. Common transformations include translations, reflections, stretches, and compressions. In the context of this question, applying the absolute value function to ƒ₁(x) = √x to create ƒ₂(x) = √|x| results in a reflection of the left side of the graph of ƒ₁ across the x-axis, thus modifying its overall shape and symmetry.
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Intro to Transformations
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Shifting and Scaling Graphs


Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.


e. Stretch vertically by a factor of 5

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Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x)        ƒ₂(x)

 x²          |x|²

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Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.


ƒ₁(x)        ƒ₂(x)

4 - x²       |4 - x²|

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Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.


ƒ₁(x)        ƒ₂(x)

 x³          |x³|

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Shifting and Scaling Graphs


Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.


f. Compress horizontally by a factor of 5

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For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.


cos 2θ + cos θ = 0

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