increasing function

Examples of increasing function in the following topics:

• Increasing, Decreasing, and Constant Functions

  • Functions can either be constant, increasing as $x$ increases, or decreasing as $x $ increases.
  • We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
  • Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval.
  • In terms of a linear function $f(x)=mx+b$, if $m$ is positive, the function is increasing, if $m$ is negative, it is decreasing, and if $m$ is zero, the function is a constant function.
  • Identify whether a function is increasing, decreasing, constant, or none of these

• What is a Quadratic Function?

  • A quadratic function is of the general form:
  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope).
  • All quadratic functions both increase and decrease.
  • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

• Relative Minima and Maxima

  • While some functions are increasing (or decreasing) over their entire domain, many others are not.  
  • A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a relative maximum.  
  • Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a relative minimum.
  • A function is also neither increasing nor decreasing at extrema.
  • This line increases towards infinity and decreases towards negative infinity, and has no relative extrema.

• Limited Growth

  • This is because exponential functions are ever-increasing.
  • As the number of resources is not increasing without bound, so too, will the human population not increase without bound.
  • Graphically, the logistic function resembles an exponential function followed by a logarithmic function that approaches a horizontal asymptote.
  • Logistic functions have an "s" shape, where the function starts from a certain point, increases, and then approaches an upper asymptote.
  • Three projections for the world's population are shown, with a low estimate reaching a peak and then decreasing, a medium estimate increasing but at an ever-slower rate, and a high estimate continuing to increase exponentially.

• Stretching and Shrinking

  • Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase.
  • If $b$ is greater than one the function will undergo vertical stretching, and if $b$ is less than one the function will undergo vertical shrinking.
  • If we want to vertically stretch the function by a factor of three, then the new function becomes:
  • Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase.
  • If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.

• Introduction to Inverse Functions

  • Below is a mapping of function $f(x)$  and its inverse function, $f^-1(x)$.  
  • This is equivalent to reflecting the graph across the line $y=x$, an increasing diagonal line through the origin.
  • In general, given a function, how do you find its inverse function?
  • Since the function $f(x)=3x^2-1$ has multiple outputs, its inverse is actually NOT a function.  
  • A function's inverse may not always be a function as illustrated above.  

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$, where $e$ is the number (approximately 2.718281828) described previously.
  • The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases.
  • $y=e^x$ is the only function with this property.
  • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The the tangent function can be graphed by plotting points as we did for the sine and cosine functions.
  • We can identify that the values of $y$ are increasing as $x$ increases and approaches $\frac{\pi}{2}$.
  • As with the sine and cosine functions, tangent is a periodic function; its values repeat at regular intervals.
  • The graph of the tangent function is symmetric around the origin, and thus is an odd function.

• Graphs of Logarithmic Functions

  • The logarithmic graph begins with a steep climb after $x=0$, but stretches more and more horizontally, its slope ever-decreasing as $x$ increases.
  • The range of the function is all real numbers.
  • At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function.
  • When graphing without a calculator we use the fact that the inverse of a logarithmic function is an exponential function.
  • The graph of the logarithmic function with base 3 can be generated using the function's inverse.

• Basics of Graphing Exponential Functions

  • The exponential function y=b^x where $b>0$ is a function that will remain proportional to its original value when it grows or decays.
  • At the most basic level, an exponential function is a function in which the variable appears in the exponent.
  • The most basic exponential function is a function of the form $y=b^x$ where $b$ is a positive number.
  • Let us consider the function $y=2^x$.
  • As you connect the points you will notice a smooth curve that crosses the y-axis at the point $(0,1)$ and is increasing as $x$ takes on larger and larger values.