<p>For which of the following functions is it true that -f(x)=f(-x) for all values of x?</p>
<p>a) f(x)= x+4
b) f(x)= x^2+4
c) f(x)= x^3+4
d) f(x)= x^2+x
e) f(x)= x^3+x</p>
<p>I know the answer is E, but I do not know how to come about this answer.</p>
<p>1) change the function to negative sign: -f(x)=-x^3-x
2) then substitute -x in place of x:
F(-x)=-x^3-x</p>
<p>As both of the functions are correct only for the equation E, E is the answer.</p>
<p>^whatever Sanjay said </p>
<p>or </p>
<p>if you take an advanced math course:
You should know that an odd function is defined by f(-x)=-f(x).
Any function with all odd powers (in this case 1 (x) and 3 (x^) is an odd function and thus, E is an odd function and fits -f(x)=f(-x).</p>
<p>Note: A is not an odd function because the number 4 (which is a constant), is treated as an even power (0). Therefore, a is false.</p>
<p>This is just a shortcut you can use if you understand it or just employ an easier method.</p>
<p>the easiest way is to plug in an number… let’s say 3…
a) -3+4 = 1 … 3+4= 7<em>-1=-7 … 1 not equal -7
b) f(x)= (-3)^2+4= 13 (3)^2+4= 13</em>-1=-13
c) f(x)= (-3)^3+4= -23 (3)^3+4 = 31<em>-1= -31
d) f(x)= (-3)^2-3 = 6 (3)^2+3= 12</em>-1=-12
e) f(x)= (-3)^3-3 = -30 (3)^3+3= 30*-1=-30
so (E) is the correct one</p>
<p>Best solution is to recognize that E) f(x) = x^3 + x is the only odd function (which is what the question is asking for).</p>
<p>Here is some technical info about odd and even functions for advanced students:</p>
<p>A function f with the property that f(-x) = -f(x) for all x is called an odd function. </p>
<p>A function f with the property that f(-x) = f(x) is called an even function.</p>
<p>Each function given in the answer choices is called a polynomial function. These are functions where each term has the form ax^n where a is a real number and n is a positive integer. </p>
<ul>
<li>Polynomial functions with only odd powers of x are odd functions. Keep in mind that x is the same as x^1, and so x is an odd power of x. From this observation we can see immediately that answer choice (E) is an odd function and thus satisfies the given definition.</li>
</ul>
<p>Polynomial functions with only even powers of x are even functions. Keep in mind that a constant c is the same as cx^0, and so c is an even power of x. For example 4 is an even power of x. From this observation we can see that the function in answer choice (B) is an even function.</p>
<p>The functions in answer choices (A), (C) and (D) have both even and odd powers of x. They are therefore functions which are neither even nor odd.</p>
<p>A quick graphical analysis of even and odd functions: The graph of an odd function is symmetrical with respect to the origin. This means that if you rotate the graph 180 degrees (or equivalently, turn it upside down) it will look the same as it did right side up. If you put choice (E) into your graphing calculator, you will see that this graph has this property.</p>
<p>Similarly, the graph of an even function is symmetrical with respect to the y-axis. This means that the y-axis acts like a “mirror,” and the graph “reflects” across this mirror. If you put choice (B) into your graphing calculator, you will see that this graph has this property. </p>
<p>Put the other three answer choices in your graphing calculator and observe that they have neither of these symmetries.</p>
<p>So another way to determine if f(-x) = -f(x) is to graph f in your graphing calculator, and see if it looks the same upside down. And another way to determine if f(-x) = f(x) is to graph f in your graphing calculator, and see if the y-axis acts like a mirror. This technique will work for all functions (not just polynomials).</p>