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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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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 1, Problema 1.1.54
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x/(x² − 1)
Guida verificata passo dopo passo1
To determine if a function is even, odd, or neither, we need to evaluate the function at -x and compare it to the original function g(x).
Calculate g(-x) by substituting -x into the function: g(-x) = (-x)/((-x)² - 1).
Simplify g(-x): Since (-x)² = x², we have g(-x) = -x/(x² - 1).
Compare g(-x) with g(x): g(x) = x/(x² - 1) and g(-x) = -x/(x² - 1). Notice that g(-x) = -g(x).
Since g(-x) = -g(x), the function g(x) is odd. A function is odd if g(-x) = -g(x) for all x in the domain of the function.
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Even Functions
A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. A common example of an even function is f(x) = x², where substituting -x yields the same output as substituting x.
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Exponential Functions
Odd Functions
A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of an odd function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, where substituting -x results in the negative of the output for x.
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Percorso guidato
06:21Properties of Functions
Function Analysis
Analyzing a function involves evaluating its behavior under transformations, such as substituting -x for x. This process helps determine whether the function is even, odd, or neither. For the function g(x) = x/(x² - 1), one must compute g(-x) and compare it to g(x) and -g(x) to classify its symmetry.
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06:29Derivatives Applied To Velocity
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