The composition of functions is an operation that combines two functions to create a new function. Given two functions, f(x) and g(x), the composition is denoted as f(g(x)), where the output of g(x) becomes the input of f(x). In other words, you apply g(x) first, and then take the result and plug it into f(x). This process allows you to chain functions together, effectively transforming the input through multiple stages. The resulting composition f(g(x)) represents a new function that embodies the combined effects of both f(x) and g(x) on the input x.
The independent variable in a function is the input value or variable that is freely chosen or controlled by the experimenter or the person using the function. It is the variable whose value is not dependent on any other variable in the context of the function. The function's rule or formula operates on the independent variable to produce the corresponding output or dependent variable.
A function f is considered odd if, for every value of x in its domain, the function's output is the negative of the function's value when the input is replaced with its negation, resulting in f (−x)=−f(x).
The inverse of a function or relation is like a "reverse" function that takes the output of the original function and tells you what input was used to get that output. It helps you "undo" the original function's operation and get back to the original value you started with. To have an inverse, each output of the original function must come from a unique input, so you can trace back to the original value.
A piecewise function is defined by multiple sub-functions, each applying to a specific interval or set of values within the domain, allowing for different behaviors in different regions.
To find the value of g(2), you need to substitute 2x=2 into the expression for g(x):
g(x)=x^4+3.
So, g (2)=2^4+3=16+3=19
Therefore, g(2) is equal to 19.
The range of a function or relation is the set of all possible output values that the function can produce for the given inputs. It represents the collection of y-values or dependent variables within the function's domain.
The domain of a function or relation is the set of all possible input values (independent variables) for which the function or relation is defined and provides a meaningful output (dependent variable). In other words, it represents all the x-values that can be used as inputs to the function or relation, ensuring that the function is well-defined and yields a unique output for each input within its domain.
If a function is not defined for a particular input in its domain, it is simply considered "undefined" for that specific input. This often occurs when certain mathematical operations, like division by zero or the square root of a negative number, are not well-defined. In such cases, the function is undefined for those particular input values, but it is still considered a valid function for the rest of its domain.
A periodic function is one that exhibits a repeating pattern at regular intervals on its graph as you move along the x-axis. The pattern repeats itself in a predictable manner, showing the same behavior at fixed intervals. Common examples of periodic functions include trigonometric functions like sine and cosine, which display wave-like patterns that continue indefinitely.
A one-to-one function is a type of function where each output value (in the range) is paired with only one unique input value (in the domain). It's like a special machine that takes different items and gives you a unique result for each item. In simpler terms, a one-to-one function never repeats its output values, ensuring that each input has a distinct and separate output.
The vertical line test determines functionality. If any vertical line intersects the graph at more than one point, it's not a function; if every vertical line intersects at most once, it is a function, as each input has a unique output (y-value).
A defined function is a mathematical operation or rule that takes an input (independent variable) and gives you a specific output. It's like a machine that takes something you put in and produces a result based on what you gave it. With a defined function, you always know what to expect as the answer for any particular input you provide.
The horizontal line test determines whether a function is one-to-one by checking if any horizontal line intersects the graph at more than one point (not one-to-one) or at most once (one-to-one).
The dependent variable in a function is the output or result that is determined by the value of the independent variable (input). It is the variable that depends on the choice of the independent variable, as the function's rule or definition calculates or maps the input to a unique output value
A function f is considered even if, for every value of x in its domain, the function's output remains the same when the input is replaced with its negation, resulting in f (−x )= f (x). This property implies that the function has symmetry about the y-axis, meaning its graph remains unchanged when reflected across the y-axis.
