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Chapter 5: Problem 30

\(f(x)=(x-a)^{2 / 5}+1\)

Short Answer

Expert verified
The domain is all real numbers: \[(-\infty, \infty)\]. The function is a horizontally shifted and vertically translated even root curve.

Step by step solution

01

- Understand the Function

Identify the different components in the function given: \[ f(x)=(x-a)^{2/5}+1 \]. Here, \(f(x)\) is a combination of a power function and a constant.
02

- Determine the Domain

The domain of a function is the set of all possible input values (\(x\)). Since the expression \((x-a)^{2/5}\) involves a root, consider that any real number raised to the power of \(2/5\) is defined. Hence, the domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \].
03

- Analyze the Shape and Transformation

Observe that \((x-a)^{2/5}\) is an even root, so it will have a similar shape to the function \(x^{2/5}\), which is a continuous curve and gradually increasing. This transformation by the value \(a\) means the curve is shifted horizontally by \(a\). Adding 1 vertically shifts the entire graph by 1 unit upwards.
04

- Sketch the Graph

To sketch the graph, start by plotting the base function \((x-a)^{2/5}\). Then, shift every point on the graph 1 unit upwards. Mark the key points when \(x=a\) as the shift doesn't change the base shape but translates it's position.
05

- Check for Symmetry

Check the symmetry of the function. Since it is an even-powered root, the function \((x-a)^{2/5}\) is symmetric about x=a. Adding a constant does not affect the symmetry relative to vertical translation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range of Functions
Understanding the domain and range of a function is crucial. The domain is the set of all possible input values (x-values) that the function can accept. For our function, \[ f(x) = (x - a)^{2/5} + 1 \], the domain is quite broad. Since any real number can be raised to the power of \(2/5\), the domain covers all real numbers. So, the domain is: \[ (-\infty, \infty) \].
Graph Transformations
Graph transformations involve moving or changing the shape of a graph. In our example, the function \[ f(x) = (x - a)^{2/5} + 1\] sees several transformations. First, \(x - a\) indicates a horizontal shift by \(a\) units to the right. This is because subtracting \(a\) from \(x\) effectively moves every point on the graph to the right.
Symmetry in Functions
Symmetry in functions helps to understand their shape and behavior better. For the function \[ f(x) = (x - a)^{2/5} + 1 \], the term \((x - a)^{2/5}\) involves an even root. Even roots exhibit symmetrical behavior. This means our base function \((x-a)^{2/5}\) is symmetric with respect to \(x = a\). Adding a constant, in this case 1, does not affect this symmetry.

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Most popular questions from this chapter

\(f(x)=(x+1)^{3}(x-2)^{2}\)

A direct current generator has an electromotive force of \(E\) volts and an internal resistance of \(r\) ohms. \(E\) and \(r\) are constants. If \(R\) ohms is the external resistance, the total resistance is \((r+R)\) ohms and if \(P\) watts is the power, then $$ P=\frac{E^{2} R}{(r+R)^{2}} $$ What external resistance will consume the most power?

If \(f(x)=a x^{3}+b x^{2}\), determine \(a\) and \(b\) so that the graph of \(f\) will have a point of inflection at \((1,2)\).

\(G(x)=\frac{2 x}{\left(x^{2}+4\right)^{3 / 2}}\)

\(f(x)=2-(x-1)^{1 / 3}\)
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