Functions are mathematical relationships represented by the letter f and the input value x, known as f(x). They may appear complex, using different algebraic symbols, but can be categorized into two types: composite and inverse.

Composite functions are formed by combining two or more functions to create a new one. This is also referred to as a function of a function. For example, fg(x) means that you find the output of g(x) and use it as the input for f(x). So, if f(x) = x + 2 and g(x) = 3x - 1, to find fg(4), first solve g(4) which equals 11. Then, insert this output into f(x) to find fg(4), which is 13. It is crucial to follow the specific order, as gf(x) would result in a different answer of 17.

An inverse function is the opposite of the original function, denoted as f⁻¹(x). It maps the outputs back to the inputs, making it a one-to-one function. On a graph, the inverse function's line will reflect the original function's line. For example, if f(x) = 2x + 4, then f⁻¹(x) = (y-4)/2.

Mappings are used to transform an input to an output. If each input has a distinct output, it can be considered a function. There are four ways to map inputs and outputs:

**Domain:**the possible inputs for the mapping**Range:**all possible outputs for the mapping**Inverse Mapping:**takes the output and maps it back to the input**Composition Mapping:**combines two or more mappings to create a new one

Graphs are visual representations of functions, with each function having a unique graph. Different factors, such as the function's equation and positivity or negativity, can influence the graph.

Polynomials are expressions containing variables raised to a positive power, multiplied by a coefficient. Though they may seem complex, they can also be simple, like x² or 3x. These expressions can be graphed to visualize the function.

Inequalities are algebraic expressions comparing two terms to show their relation, with symbols like >, <, ≥, and ≤. For instance, 2x > 4 means 2x is greater than 4.

- Functions involve an input that affects the output.

- Functions are represented as f(x) using algebraic symbols.

- There are two types of functions: composite and inverse.

- Mappings are used to show the domain and range of a function.

In conclusion, functions are essential mathematical concepts that have various real-life applications. By understanding the relationship between inputs and outputs and knowing how to solve composite and inverse functions, you can use functions to solve complicated problems. So, the next time you encounter a function, don't be daunted - remember these key takeaways, and you'll be able to approach it with confidence.

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