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h(x) = 1 (x-2)3

h(x) = 1 (x-2)3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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find functions f(x) and g(x) so given function can be expressed as h(x)=f(g(x))

The image shows a mathematical function defined as: \( h(x) = \frac{1}{(x-2)^3} \) This is a rational function where the denominator is \((x-2)^3\). The function has a vertical asymptote at \(x = 2\) because the denominator becomes zero at this point, leading to an undefined value for \(h(x)\). As \(x\) approaches 2 from either side, the function tends toward positive or negative infinity, depending on the direction of approach. This behavior can be important for understanding the limits and continuity of such functions.
Transcribed Image Text:The image shows a mathematical function defined as: \( h(x) = \frac{1}{(x-2)^3} \) This is a rational function where the denominator is \((x-2)^3\). The function has a vertical asymptote at \(x = 2\) because the denominator becomes zero at this point, leading to an undefined value for \(h(x)\). As \(x\) approaches 2 from either side, the function tends toward positive or negative infinity, depending on the direction of approach. This behavior can be important for understanding the limits and continuity of such functions.
The image displays a mathematical function: \[ h(x) = (x + 2)^2 \] This represents a quadratic function where the expression \( (x + 2) \) is squared. The graph of this function would be a parabola opening upwards with its vertex at the point \((-2, 0)\) on the Cartesian plane. The transformation \(x + 2\) indicates a horizontal shift of the standard parabola \(y = x^2\) to the left by 2 units.
Transcribed Image Text:The image displays a mathematical function: \[ h(x) = (x + 2)^2 \] This represents a quadratic function where the expression \( (x + 2) \) is squared. The graph of this function would be a parabola opening upwards with its vertex at the point \((-2, 0)\) on the Cartesian plane. The transformation \(x + 2\) indicates a horizontal shift of the standard parabola \(y = x^2\) to the left by 2 units.
Expert Solution
Step 1

(a) consider functions

       f(x) = 1/x3 and g(x) = x - 2.

      Then, f(g(x)) = f(x - 2) 

                             = 1/(x - 2)3 = h(x) 

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