Functions are essential tools in mathematical modelling, allowing us to map inputs from a domain to outputs within a range. When analysing how functions behave at extreme values, mathematicians look at whether the outputs are confined within specific limits. Understanding whether a function is constrained or can grow infinitely is a key aspect of A-Level Mathematics and graph theory.
Theory
A function is described as bounded if its outputs are restricted within certain barriers on a coordinate grid. We categorise these boundaries based on whether they limit the function from above, from below, or both.
Bounded from Above
A function f(x) is bounded from above if its outputs can never exceed a specific real number, denoted as M. This number M is called an upper bound.
Formal definition:
for all elements x in the domain.
In terms of graphical behaviour, you can draw a horizontal line y = M such that the entire curve lies on or below it. For instance, a negative quadratic graph has a maximum turning point that serves as its sharpest upper bound.
Bounded from Below
A function f(x) is bounded from below if its outputs can never drop below a specific real number, denoted as m. This number m is called a lower bound.
Formal definition:
for all elements x in the domain.
Graphically, this means you can draw a horizontal line y = m such that the entire curve lies on or above it. A standard positive quadratic graph has a minimum vertex that acts as its lower bound.
Bounded Functions
A function is classified as a fully bounded function if it is bounded both from above and from below. This means the entire range of the function is trapped within a specific horizontal interval.
Formal definition:
for all elements x in the domain.
On a graph, the complete curve stays confined inside a horizontal strip bounded by y = m and y = M. Trigonometric parent functions like sine and cosine are classic examples of fully bounded functions.
Worked Example
Problem: Find the maximum or minimum turning points to determine the bounds of the function:
for all real values of x, and state its boundedness status.
Step-by-step Solution:
Rearrange the quadratic expression to prepare for completing the square:
Factor out the negative sign from the variable terms:
Complete the square inside the bracket:
Expand the brackets and simplify the constants:
Analyse the completed square form to find the bounds. Since any real number squared must be greater than or equal to zero, the term:
This means that:
The maximum possible output occurs when the squared term equals zero, which yields f(3) = 16. For any other value of x, the function output will be less than 16.
Conclusion: The function is bounded from above with an upper bound of 16. It is unbounded from below because as x approaches infinity, the output approaches negative infinity.
Practice Questions & Solutions
Find the lower bound of the following function for all real values of x:
First, complete the square for the given quadratic function:
Since any real squared number cannot be negative:
Adding 3 to both sides of the inequality demonstrates that:
Therefore, the function is bounded below with a sharp lower bound of:
Determine the exact upper and lower bounds for the following function over all real inputs:
Start with the standard boundary interval for a cosine function:
Multiply each part of the inequality chain by negative 4 and reverse the inequality signs:
Rearrange the inequality back to standard numerical order:
Add 5 to all parts of the inequality to match the original function:
Hence, the function is fully bounded with a lower bound of 1 and an upper bound of 9.
Find the upper and lower bounds of the given function across the restricted domain:
For the domain:
Because this reciprocal function is strictly decreasing for positive values of x, evaluate the outputs at the boundaries. Calculate the value at the upper domain limit:
Calculate the value at the lower domain limit:
This establishes the bounded range of the function on this interval as:
The lower bound is:
The upper bound is:
Determine the bounds for the given exponential function over its specified domain:
For the domain:
As x grows infinitely large, the exponential term approaches zero from above:
This establishes a strict horizontal asymptote where the output values are greater than 2: f(x) > 2
Evaluate the function at the starting boundary where x equals 0:
This shows that the function values are contained within the following interval:
Therefore, the lower bound is 2 and the upper bound is 3.
Find the upper bound of the following function defined for all real numbers:
A squared real number must be greater than or equal to zero:
Multiplying by negative 3 reverses the inequality sign:
Add 12 to both sides of the inequality to reconstruct the function:
The function can never exceed this value, meaning the upper bound is:
Summarise with AI:
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