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Chapter 7: Problem 56

Graph each function, and give its domain and range. $$ f(x)=\sqrt[3]{x}+1 $$

Short Answer

Expert verified
Domain: \((-\infty, +\infty)\) Range: \((-\infty, +\infty)\)

Step by step solution

01

- Identify the Function

Recognize that the given function is a cube root function combined with a vertical shift. The general form is given as: \[ f(x)=\root[ 3 ]{x} + 1 \]
02

- Determine the Domain

For any cube root function, there are no restrictions on the value of \(x\). Thus, the domain of the function is all real numbers. \[ \text{Domain: } (-\infty, +\infty) \]
03

- Determine the Range

For a cube root function, as \( x \) ranges over all real numbers, \( \root[3]{x} \) also ranges over all real numbers. Since we added 1 to every value, the range remains all real numbers. \[ \text{Range: } (-\infty, +\infty) \]
04

- Plot Key Points

Choose key points to plot the function. For instance:- When \( x = 0, f(x) = \root[3]{0} + 1 = 1 \)- When \( x = 1, f(x) = \root[3]{1} + 1 = 2 \)- When \( x = -1, f(x) = \root[3]{-1} + 1 = 0 \)- When \( x = 8, f(x) = \root[3]{8} + 1 = 3 \)- When \( x = -8 , f(x) = \root[3]{-8} + 1 = -1 \)
05

- Sketch the Graph

Plot the points from Step 4 on a coordinate plane: (0, 1), (1, 2), (-1, 0), (8, 3), (-8, -1). Draw a smooth curve that passes through these points and extends in both directions.

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Key Concepts

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

Domain and Range
The domain of a function provides all possible input values, while the range offers all possible output values. For the cube root function combined with a vertical shift, we consider the function: \( f(x) = \root[3]{x} + 1 \). Since there's no restriction on the value of \ x\ in cube root functions, the domain is all real numbers, written as \( (-\infty, +\infty) \). The same applies to the range because every number we plug into the cube root function will produce a real number. The output of \( \root[3]{x} + 1 \) is also unrestricted because adding 1 adjusts the range but does not limit it. Therefore, the range is also all real numbers, written as \( (-\infty, +\infty) \).
Vertical Shift
Vertical shifts occur when a constant is added or subtracted from a function. In our case, \( f(x) = \root[3]{x} + 1 \), the \ + 1 \ means the entire cube root function shifts up one unit on the y-axis. This means every output value is 1 unit higher than the unmodified cube root function \( \root[3]{x} \). Visualizing this, imagine the standard \( \root[3]{x} \) graph and simply lift each point 1 unit up. If originally, \( \root[3]{8} = 2 \), now \( \root[3]{8} + 1 = 3 \).
Plotting Points
Plotting points allows you to see the transformation of the function visually. You pick a few key values of \ x \ and compute \ f(x) \ to get coordinate pairs, then plot these on a graph. Let's choose \( x = 0, 1, -1, 8, -8 \). For \( x = 0 \, \ f(x) = \root[3]{0} + 1 = 1 \). For \( x = 1 \, \ f(x) = \root[3]{1} + 1 = 2 \. For \ x = -1 \, \ f(x) = \root[3]{-1} + 1 = 0 \. For \ x = 8 \, \ f(x) = \root[3]{8} + 1 = 3 \). And for \ x = -8 \, \ f(x) = \root[3]{-8} + 1 = -1 \ . With these points: \ (0, 1), (1, 2), (-1, 0), (8, 3), (-8, -1) \, we can draw a smooth curve that passes through these points and extends both directions.

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