As a starting point for function analysis, let’s just list characteristics of our Core Functions. A full analysis would also provide reasoning.
Domain: The domain \((-\infty , \infty )\).
Zeros: Core Exponential Functions have no zeros.
Continuity: Core Exponential Functions are continuous.
End-Behavior:
If \(r > 1\), the Core Exponential Functions tend to \(0\) as the domain becomes unbounded negatively. Core Exponential Functions tend to \(\infty \) as the domain becomes unbounded positively.
If \(0 < r < 1\), then this is reversed.
Behavior (Inc/Dec):
If \(r > 1\), then Core Exponential Functions are increasing functions.
If \(0 < r < 1\), then Core Exponential Functions are decreasing functions.
Global Maximum and Minimum: Core Exponential Functions have no global maximum or minimum.
Local Maximum and Minimum: Core Exponential Functions have no local maximums or minimums.
Range: The range of a Core Exponential Function is either \((0,\infty )\) or \((-\infty , 0)\).
Domain: The domain \((0, \infty )\).
Zeros: Core Logarithmic Function have \(1\) as their only zero.
Continuity: Core Logarithmic Functions are continuous.
End-Behavior: Since the domain is \((0, \infty )\), Core Logarithmic Functions only end-behavior as the domain become unbounded positively.
If \(r > 1\), then
The other side is called Singularity Behavior.
If \(0 < r < 1\), then
The other side is called Singularity Behavior.
Behavior (Inc/Dec):
If \(r > 1\), then Core Logarithmic Functions are increasing functions.
If \(0 < r < 1\), then Core Logarithmic Functions are decreasing functions.
Global Maximum and Minimum: Core Logarithmic Functions have no global maximum or minimum.
Local Maximum and Minimum: Core Logarithmic Functions have no local maximums or minimums.
Range: The range of Core Logarithmic Functions is \((-\infty ,\infty )\).
Domain: The domain both functions is \((-\infty , \infty )\).
Zeros: Both functions have zeros and since both functions are periodic, they have an infinite number of zeros.
- Zeros of \(\sin (x)\) are \(k \pi \, | \, k \in \mathbb {Z}\).
- Zeros of \(\cos (x)\) are \(\frac {\pi }{2} + k \pi \, | \, k \in \mathbb {Z}\).
Continuity: Both functions are continuous.
End-Behavior: Both functions continue to oscillate.
Behavior (Inc/Dec): Both functions switch back-and-forth between increasing and decreasing with a period of \(2\pi \).
- \(\sin (x)\) increases on \(\left (0, \frac {\pi }{2} \right )\)
- \(\sin (x)\) decreases on \(\left (\frac {\pi }{2}, \frac {3\pi }{2} \right )\)
- \(\sin (x)\) increases on \(\left (\frac {3\pi }{2}, 2\pi \right )\)
- \(\cos (x)\) decreases on \((0, \pi )\)
- \(\cos (x)\) increases on \((\pi . 2\pi )\)
Global Maximum and Minimum: Both functions have a global maximum of \(1\) and a global minimum of \(-1\).
Local Maximum and Minimum: Both functions \(1\) and \(-1\) as their local extrema.
Range: The range of both functions is \([-1, 1]\).
Domain: The domain is \((-\infty , \infty )\).
Zeros: \(0\) is the only zero.
Continuity: The Core Absolute Value Function is continuous.
End-Behavior: The Core Absolute Value Function becomes unbounded positively on both sides.
Behavior (Inc/Dec):
The Core Absolute Value Function decreases on \((-\infty , 0)\) and increases on \((0, \infty )\)
Global Maximum and Minimum: The Core Absolute Value Function has \(0\) as its global minimum and it has no global maximum.
Local Maximum and Minimum: The Core Absolute Value Function has \(0\) as its only local minimum and it has no local maximum.
Range: The range is \([0, \infty )\).
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more examples can be found by following this link
More Examples of Visual Behavior