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Chapter 7: Problem 12

A power function is given. Evaluate the function at the indicated value, then graph the function for the specified independent variable values. Round the function values to two decimal places as necessary. \(f(x)=x^{0.7} ;\) Evaluate \(f(0), f(10), f(20) .\) Graph \(f(x)\) for \(0 \leq x \leq 30\)

Short Answer

Expert verified
Evaluated values are: \(f(0) = 0\), \(f(10) \approx 5.01\), \(f(20) \approx 9.54\). Graph the function for \(0 \leq x \leq 30\).

Step by step solution

01

Evaluate the function at x=0

The given function is \(f(x) = x^{0.7}\). We need to find \(f(0)\). By substituting \(x = 0\), we have: \[ f(0) = 0^{0.7} = 0 \] because any non-zero number raised to any power is 0.
02

Evaluate the function at x=10

Next, we evaluate \(f(x)\) at \(x = 10\). Substitute \(x = 10\) into the function: \[ f(10) = 10^{0.7} \] Using a calculator, we calculate \(10^{0.7} \approx 5.01\) when rounded to two decimal places.
03

Evaluate the function at x=20

Now evaluate \(f(x)\) at \(x = 20\). Substitute \(x = 20\) into the function: \[ f(20) = 20^{0.7} \] Using a calculator, we find that \(20^{0.7} \approx 9.54\) when rounded to two decimal places.
04

Graph the function for 0 鈮� x 鈮� 30

To graph \(f(x) = x^{0.7}\) for the range \(0 \leq x \leq 30\), create a table of values by choosing several \(x\)-values, such as 0, 5, 10, 15, 20, 25, and 30, and evaluate \(f(x)\) for these points. Plot these points on a graph, and draw a smooth curve through them. The graph should show a gradual increase as \(x\) increases, having already evaluated \(f(0)\), \(f(10)\), and \(f(20)\). The curve should start at the origin (0,0) and increase as \(x\) increases.

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Key Concepts

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

Function Evaluation
Evaluating a function means finding its value at a given point. It's like checking how a machine works by feeding it specific inputs. In our case, the function is a power function given as \( f(x) = x^{0.7} \). You evaluate it by substituting the input value for \( x \) in the equation. For example:
  • For \( f(0) \), substitute \( 0 \) into the function: \( 0^{0.7} = 0 \). Any number raised to a power is zero, hence the result is 0.
  • For \( f(10) \), substitute \( 10 \): \( 10^{0.7} \approx 5.01 \). A calculator helps here, especially for non-integer powers.
  • Similarly, for \( f(20) \): \( 20^{0.7} \approx 9.54 \).
Evaluating these values gives a snapshot of the function's behavior at those specific points.
Graphing Functions
Graphing a function paints a picture of its behavior across a range of values. It offers a visual representation, which is often more intuitive.
To graph \( f(x) = x^{0.7} \) over \( 0 \leq x \leq 30 \):
  • Create a table of values. Choose a few key \( x \)-values like 0, 5, 10, 15, 20, 25, and 30.
  • Evaluate the function at these \( x \)-values to get corresponding \( f(x) \) values.
  • For instance, at \( x = 10 \), \( f(x) \approx 5.01 \) and at \( x = 20 \), \( f(x) \approx 9.54 \).
  • Plot these points on a graph. Each \( x \) corresponds to a height, the \( f(x) \), above the x-axis.
  • Draw a smooth curve through the points. The graph should start from the origin \( (0, 0) \) and rise gradually. The biggest leap is near the start, slowing as \( x \) grows.
Graphing allows us to observe trends and patterns in the function beyond the numbers.
Exponentiation
Exponentiation is a key mathematical operation. It involves raising a number, known as the base, to a power (exponent). It's a way to express repeated multiplication.
For instance, in \( 10^{0.7} \):
  • \( 10 \) is the base, and \( 0.7 \) is the exponent.
  • This does not mean multiplying 10 by itself. Instead, a power of less than 1 implies a root or fractional power.For example, \( 10^{0.5} \) is the square root of 10.
  • At a decimal exponent like 0.7, calculations involve roots, requiring calculators to find precise values.
Exponentiation expands our toolkit to represent and compute larger numbers or those in between traditional intervals. It's crucial for understanding how power functions behave across different inputs.

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Most popular questions from this chapter

A natural logarithm function is given. Evaluate the function at the indicated values, then graph the function for the specified independent variable values. Round the function values to three decimal places as necessary. $$f(x)=12.3+1.2 \cdot \ln x ; \text { Evaluate } f(0.1), f(2), f(3) . \text { Graph } f(x) \text { for } 0.1 \leq x \leq 3$$

A natural exponential function is given. Evaluate the function at the indicated values, then graph the function for the specified independent variable values. Round the function values to three decimal places as necessary. $$f(x)=25 e^{-0.25 x} ; \text { Evaluate } f(0), f(6), f(10) . \text { Graph } f(x) \text { for } 0 \leq x \leq 10$$

Consider the following information: The amount of carbon monoxide in a cigarette based on its tar and nicotine content and its weight can be estimated by $$y=3.2+0.96 x_{1}-2.63 x_{2}-0.13 x_{3}$$ Evaluate and interpret \(y\) when \(x_{1}=14.1, x_{2}=0.86, x_{3}=0.9853\)

A power function is given. Evaluate the function at the indicated value, then graph the function for the specified independent variable values. Round the function values to two decimal places as necessary. \(f(x)=x^{1.3} ;\) Evaluate \(f(0), f(5), f(10) .\) Graph \(f(x)\) for \(0 \leq x \leq 20\)

Consider the following information. Round the answers to one decimal place: A naturalist takes a sample of sturgeons and compares their length to their weight and determines that they can be modeled by $$ f(x)=27.7 e^{0.008 x} \quad 250 \leq x \leq 500 $$ where \(x\) represents the length of the fish in millimeters \((\mathrm{mm})\) and \(f(x)\) represents the weight in grams (g). What is the estimated weight of a sturgeon that is \(300 \mathrm{mm}\) long?
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