Functions are a fundamental topic in H2 Math, and it is essential for junior college students to have a strong understanding of this topic in order to excel in their A-level exams.
Topical Revision for all the subtopics in Functions H2 Math For Junior College Students
In this article, we will be revisiting the various subtopics included in the Functions syllabus, providing a comprehensive topical revision to help students reinforce their understanding of this important topic.
Relations & Functions
A relation is a set of ordered pairs, where each pair has an x-value and a y-value. A function is a specific type of relation in which each x-value corresponds to exactly one y-value. To determine if a relation is a function, the vertical line test can be used. This test states that if a vertical line intersects the relation in more than one point, then the relation is not a function.
Representing Functions
Functions can be represented in several ways, including algebraically, graphically, and verbally. Algebraically, a function can be represented using an equation in the form y = f(x), where x is the independent variable and y is the dependent variable. Graphically, functions can be represented as a set of points on a coordinate plane, with x and y values plotted to produce a graph. Verbal representations of functions describe the relationship between the input and output values, such as “f(x) represents the height of a building, given the value of x, which represents the number of floors”.
Vertical Line Test
As mentioned previously, the vertical line test is used to determine if a relation is a function. If a vertical line intersects the relation in more than one place, then the relation is not a function. This test is an important tool for students to use when determining the validity of a function, and is frequently tested in H2 Math exams.
Inverse Functions
An inverse function is a function that reverses the inputs and outputs of another function. If a function f(x) has an inverse, then its inverse, denoted as f^-1(x), will have the same range as the original function but its domain will be the range of the original function. To determine if a function has an inverse, the horizontal line test can be used. If every horizontal line intersects the function in exactly one point, then the function has an inverse.
Graphical Relationship Between A Function And Its Inverse
When a function and its inverse are graphed on the same coordinate plane, they will intersect at exactly one point, which is the point (1,1). This relationship is important for students to understand, as it is frequently tested in H2 Math exams.
Composite Functions
Composite functions are functions formed by combining two or more functions. The composite function, denoted as f(g(x)), represents the function f applied to the result of the function g. The domain of the composite function is determined by the intersection of the domains of the individual functions, and the range is determined by the range of the function f.
Domain Of Composite Functions
The domain of a composite function is the set of all values for which the composite function is defined. When combining functions to form a composite function, it is important to consider the domain of each individual function and determine the intersection of their domains to find the domain of the composite function.
Range Of Composite Functions
The range of a composite function is the set of all possible output values of the function. The range of a composite function is determined by the range of the outer function, as it is this function that determines the final output value.
