Discussing the concept of even and odd functions graphically and analytically.
Even Functions
The graph of an even function is symmetric with respect to the y-axis.
To prove analytically that a given function \(f\) is even, we need to prove that:
Odd Functions
The graph of an odd function is symmetric with respect to the origin \((0,0)\).
To prove analytically that a given function \(f\) is odd, we need to prove that:
Example 1: Analytical Proofs for Even Functions
The formulas of the even functions \(f\), \(g\), \(h\), and \(i\) are given (see graphs above). Show analytically that each of these functions satisfies the property of an even function: \(f(-x) = f(x)\).
Solutions to Example 1
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1) \(f(x) = x^2\)
\[ f(-x) = (-x)^2 = (-x)(-x) = x^2 \]
Since \(f(-x) = f(x)\), \(f(x)\) is an even function.
2) \(g(x) = -|x| + 2\)
\[ g(-x) = -|-x| + 2 \]
The absolute value function \(|x|\) is an even function, therefore \(|-x| = |x|\).
Simplifying \(g(-x)\):
\[ g(-x) = -|x| + 2 \]
Since \(g(-x) = g(x)\), \(g(x)\) is an even function.
3) \(h(x) = 10 e^{-0.2 x^2}\)
\[ h(-x) = 10 e^{-0.2 (-x)^2} = 10 e^{-0.2 x^2} \]
Since \(h(-x) = h(x)\), \(h(x)\) is an even function.
4) \(i(x) = 8 \cos(x)\)
\[ i(-x) = 8 \cos(-x) \]
The cosine function is an even function, therefore \(\cos(-x) = \cos(x)\), which gives:
\[ i(-x) = 8 \cos(-x) = 8 \cos(x) \]
Since \(i(-x) = i(x)\), \(i(x)\) is an even function.
Example 2: Analytical Proofs for Odd Functions
The formulas of the odd functions \(j\), \(k\), \(l\), and \(m\) are given (see graphs above). Show analytically that each of these functions satisfies the property of an odd function: \(f(-x) = -f(x)\).
Solutions to Example 2
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1) \(j(x) = x^3\)
\[ j(-x) = (-x)^3 = (-x)(-x)(-x) = -x^3 \]
Since \(j(-x) = -j(x)\), \(j(x)\) is an odd function.
2) \(k(x) = -x\)
\[ k(-x) = -(-x) = x \]
Since \(k(-x) = -k(x)\), \(k(x)\) is an odd function.
3) \(l(x) = 6 \sin(x)\)
\[ l(-x) = 6 \sin(-x) \]
The sine function is an odd function, therefore \(\sin(-x) = -\sin(x)\), which gives:
\[ l(-x) = 6(-\sin(x)) = -6 \sin(x) \]
Since \(l(-x) = -l(x)\), \(l(x)\) is an odd function.
4) \(m(x) = e^x - e^{-x}\)
\[ m(-x) = e^{-x} - e^{-(-x)} = e^{-x} - e^x \]
Rewriting by factoring out \(-1\):
\[ m(-x) = -(e^x - e^{-x}) \]
Since \(m(-x) = -m(x)\), \(m(x)\) is an odd function.
Exercises
Verify analytically whether each of these functions is even, odd, or neither:
- \( f(x) = -2x^2 + 4 \)
- \( g(x) = 3|x| - x^2 \)
- \( h(x) = \dfrac{1}{x} \)
- \( i(x) = \tan(x) \)
- \( j(x) = x^2 + 3x \)
Detailed Solutions to Exercises
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1. \(f(x) = -2x^2 + 4\)
Find \(f(-x) = -2(-x)^2 + 4 = -2x^2 + 4\).
Since \(f(-x) = f(x)\), function \(f\) is even.
2. \(g(x) = 3|x| - x^2\)
Find \(g(-x) = 3|-x| - (-x)^2 = 3|x| - x^2\).
Since \(g(-x) = g(x)\), function \(g\) is even.
3. \(h(x) = \dfrac{1}{x}\)
Find \(h(-x) = \dfrac{1}{-x} = -\dfrac{1}{x}\).
Since \(h(-x) = -h(x)\), function \(h\) is odd.
4. \(i(x) = \tan(x)\)
Find \(i(-x) = \tan(-x) = -\tan(x)\).
Since \(i(-x) = -i(x)\), function \(i\) is odd.
5. \(j(x) = x^2 + 3x\)
Find \(j(-x) = (-x)^2 + 3(-x) = x^2 - 3x\).
Notice that \(j(-x) \neq j(x)\) and \(j(-x) \neq -j(x)\). Therefore, \(j(x)\) is neither even nor odd.