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Chapter 8: Problem 7

(Graphing program optional.) A function is said to be even if \(f(-x)=f(x)\) and odd if \(f(-x)=-f(x)\) for all \(x\) in \(f^{\prime}\) s domain. Use these definitions to: a. Show that the even power functions are even. b. Show that the odd power functions are odd. c. Show whether each of the following functions is even, odd, or neither. i. \(f(x)=x^{4}+x^{2}\) iii. \(h(x)=x^{4}+x^{3}\) ii. \(u(x)=x^{5}+x^{3}\) iv. \(g(x)=10.3^{x}\) d. For each function that you have identified as even or odd, what do you predict about the symmetry of its graph? If possible, check your predictions with a function graphing program.

Short Answer

Expert verified
f(x) = x^4 + x^2 is even. h(x) = x^4 + x^3 is neither. u(x) = x^5 + x^3 is odd. g(x) = 10.3^x is neither.

Step by step solution

01

Definition of Even and Odd Functions

A function is defined to be even if f(x) = f(-x)and a function is defined to be odd if f(-x) = -f(x)for all x in the function's domain.
02

Show that even power functions are even

Consider the function f(x) = x^n where n is an even number. We need to show that f(-x) = (-x)^n = x^n. Since n is even, (-x)^n = x^n. Thus, f(-x) = f(x), indicating that the function is even.
03

Show that odd power functions are odd

Consider the function f(x) = x^n where n is an odd number. We need to show that f(-x) = (-x)^n = -x^n. Since n is odd, (-x)^n = -x^n. Thus, f(-x) = -f(x), indicating that the function is odd.
04

Identifying the nature of f(x) = x^4 + x^2

Evaluate f(-x): f(-x) = (-x)^4 + (-x)^2 = x^4 + x^2 = f(x). So, f(x) is an even function.
05

Identifying the nature of h(x) = x^4 + x^3

Evaluate h(-x): h(-x) = (-x)^4 + (-x)^3 = x^4 - x^3 eq h(x) and h(-x) = x^4 - x^3 eq -h(x). So, h(x) is neither even nor odd.
06

Identifying the nature of u(x) = x^5 + x^3

Evaluate u(-x): u(-x) = (-x)^5 + (-x)^3 = -x^5 - x^3 = - (x^5 + x^3) = -u(x). So, u(x) is an odd function.
07

Identifying the nature of g(x) = 10.3^x

Evaluate g(-x): g(-x) = 10.3^{-x} eq g(x) and g(-x) = 10.3^{-x} eq -g(x). So, g(x) is neither even nor odd.
08

Predicting Symmetry

For each function that has been identified as even or odd, predict its symmetry: - Even functions have symmetry about the y-axis. - Odd functions have symmetry about the origin. - f(x) = x^4 + x^2 is even and will be symmetric about the y-axis. - u(x) = x^5 + x^3 is odd and will be symmetric about the origin.
09

Checking Predictions with Graphing Program

Use a graphing program to plot the functions and check the symmetry: - Plot f(x) = x^4 + x^2 and observe the y-axis symmetry. - Plot u(x) = x^5 + x^3 and observe the origin symmetry.

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Key Concepts

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

Power Functions
Power functions have the form \( f(x) = x^n \), where \( n \) is a real number. For the topic of function symmetry, we specifically focus on integer values of \( n \). When \( n \) is an even integer, the power function is identically even. This means that \( f(-x) = f(x) \). For example, if we take \( f(x) = x^4 \), then evaluating \( f(-x) \) results in \( (-x)^4 = x^4 \), demonstrating that \( f(x) = f(-x) \).
When \( n \) is an odd integer, the power function is odd, denoting that \( f(-x) = -f(x) \). An example is \( f(x) = x^3 \). Calculating \( f(-x) \) results in \( (-x)^3 = -x^3 \) which equals \( -f(x) \). Hence, the function is odd.
Power functions lay the groundwork for understanding more complex functions that can be decomposed into sums of even and odd power functions.
Function Symmetry
Function symmetry is the property that allows us to identify whether a function is even, odd, or neither. These properties are crucial for understanding the behavior of graphs.

  • An even function follows the rule \( f(x) = f(-x) \) for all x in its domain. This means that the graph of the function is symmetric about the y-axis.

    • Example: \( f(x) = x^4 + x^2 \). Calculate \( f(-x) \): \( (-x)^4 + (-x)^2 \) simplifies to \( x^4 + x^2 \), confirming \( f(-x) = f(x) \). Hence, the function is even.
  • An odd function adheres to the rule \( f(-x) = -f(x) \) for all x in its domain. This makes the function symmetric about the origin.

    • Example: \( f(x) = x^5 + x^3 \). Calculate \( f(-x) \): \( (-x)^5 + (-x)^3 \) simplifies to \( -x^5 - x^3 \), confirming \( f(-x) = -f(x) \). Hence, the function is odd.

However, if a function does not satisfy either of these definitions, it is classified as neither even nor odd. Example: \( f(x) = x^4 + x^3 \). Calculate \( f(-x) \): \( (-x)^4 + (-x)^3 \) simplifies to \( x^4 - x^3 \), which is neither equal to \( f(x) \) nor \( -f(x) \), making the function neither even nor odd.
Graphing Functions
Graphing functions provides a visual way to understand their behavior. When dealing with even and odd functions, predicting symmetry helps us quickly sketch these graphs.
  • For even functions, symmetry about the y-axis means if you fold the graph along the y-axis, both sides will match.
    • Example: Plot \( f(x) = x^4 + x^2 \) on a graphing calculator or software and notice how the graph mirrors itself about the y-axis.
  • For odd functions, symmetry about the origin means rotating the graph 180 degrees around the origin will yield the same graph.
    • Example: Plot \( f(x) = x^5 + x^3 \) and observe its rotational symmetry around the origin.
    If a function is neither even nor odd, its graph will not display these specific symmetries.
    Example: Plot \( f(x) = x^4 + x^3 \) and observe the lack of y-axis or origin symmetry. Combining understanding from algebra and visual confirmation through graphing tools helps reinforce the concept of function symmetry.

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Most popular questions from this chapter

Market research suggests that if a particular item is priced at \(x\) dollars, then the weekly profit \(P(x)\), in thousands of dollars, is given by the function $$ P(x)=-9+\frac{11}{2} x-\frac{1}{2} x^{2} $$ a. What price range would yield a profit for this item? b. Describe what happens to the profit as the price increases. Why is a quadratic function an appropriate model for profit as a function of price? c. What price would yield a maximum profit?

For the following quadratic functions in vertex form, \(f(x)=a(x-h)^{2}+k,\) determine the values for \(a, h,\) and \(k\) Then compare each to \(f(x)=x^{2},\) and identify which constants represent a stretch/compression factor, or a shift in a particular direction. a. \(p(x)=5(x-4)^{2}-2\) b. \(g(x)=\frac{1}{3}(x+5)^{2}+4\) c. \(h(x)=-0.25\left(x-\frac{1}{2}\right)^{2}+6\) d. \(k(x)=-3(x+4)^{2}-3\)

a. If the degree of a polynomial is odd, then at least one of its zeros must be real. Explain why this is true. b. Sketch a polynomial function that has no real zeros and whose degree is: i. 2 ii. 4 c. Sketch a polynomial function of degree 3 that has exactly: i. One real zero ii. Three real zeros d. Sketch a polynomial function of degree 4 that has exactly two real zeros.

Complex number expressions can be multiplied using the distributive property or the FOIL technique. Multiply and simplify the following. ( Note: \(i^{2}=-1 .\) ) a. \((3+2 i)(-2+3 i)\) d. \((5-3 i)(5+3 i)\) b. \((4-2 i)(3+i)\) e. \((3-i)^{2}\) c. \((2+i)(2-i)\) f. \((4+5 i)^{2}\)

A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is. a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway. b. Construct an equation that represents the highway (using \(x\) for miles east and \(y\) for miles north). c. Now use the Pythagorean Theorem to describe the square of the distance, \(d,\) of the pilot to any point \((x, y)\) on the highway. d. Substitute the expression for \(y\) from part (b) into the equation from part (c) in order to write \(d^{2}\) as a quadratic in \(x\) e. If we minimize \(d^{2}\), we minimize the distance \(d\). So let \(D=d^{2}\) and write \(D\) as a quadratic function in \(x\). Now find the minimum value for \(D\). f. What are the coordinates of the closest point on the highway, and what is the distance, \(d\), to that point?
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