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

• Interest Rates and the Business Cycle

  • Consequently, the bond's supply increases.
  • Thus, the bond's demand increases and the demand function shifts rightward in Figure 8.
  • In this case, both functions increase, causing the quantity of bonds to increase, but bond prices and interest rates become unknown.
  • First, increase the demand function by a good deal, and increase the supply function by a little.
  • Second, increase the demand function by a little and increase the supply function by much.

• 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.

• Defining the Production Function

  • This is also known as diminishing returns to scale - increasing the quantity of inputs creates a less-than-proportional increase in the quantity of output.
  • Increasing marginal costs can be identified using the production function.
  • If a firm has a production function Q=F(K,L) (that is, the quantity of output (Q) is some function of capital (K) and labor (L)), then if 2Qfunction has increasing marginal costs and diminishing returns to scale.
  • From this production function we can see that this industry has constant returns to scale - that is, the amount of output will increase proportionally to any increase in the amount of inputs.
  • Finally, the Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, output will not change.

• Derivatives of Exponential Functions

  • The derivative of the exponential function is equal to the value of the function.
  • Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
  • The slope of the graph at any point is the height of the function at that point.
  • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

• The consumption function

  • The consumption function is a single mathematical function used to express consumer spending.
  • In economics, the consumption function is a single mathematical function used to express consumer spending.
  • The function is used to calculate the amount of total consumption in an economy.
  • Since it is assumed that as income increases spending increases under the MPC, this means that the slope of the consumption function is positive .
  • Since it is assumed that the consumption increases as income increases, the slope of the function is positive.

• Functional Structure

  • An organization with a functional structure is divided based on functional areas, such as IT, finance, or marketing.
  • Functional departments arguably permit greater operational efficiency because employees with shared skills and knowledge are grouped together by functions performed.
  • This arrangement allows for increased specialization.
  • Functional structures may also be susceptible to tunnel vision, with each function perceiving the organization only from within the frame of its own operation.
  • Each different functions (e.g., HR, finance, marketing) is managed from the top down via functional heads (the CFO, the CIO, various VPs, etc.).

• 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.