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A linear function is a type of mathematical function that creates a straight line when graphed. It takes the general form of \( f(x) = mx + c \), where \( m \) stands for the slope and \( c \) represents the y-intercept.
Here's why every part is important:

• The slope \( m \) tells us how steep the line is. If the slope is positive, the line rises. If it's negative, like in our example function \( f(x) = -2x + 4 \), the line falls as it moves from left to right.
• The y-intercept \( c \) is the point where the line crosses the y-axis. In our function, it crosses at \( c = 4 \).

Linear functions are straightforward; they don't curve or bend. This clarity makes them very predictable and easy to work with in many math problems.

The horizontal line test is an essential graphical method to determine if a function is one-to-one. If you can draw any horizontal line that touches the graph of the function more than once, then the function is not one-to-one. However, if every horizontal line touches the graph at most once, the function is one-to-one.
For the function \( f(x) = -2x + 4 \), since it is a straight line with a constant slope of \(-2\), no horizontal line will ever touch it more than once. This is because the function continuously decreases or increases without repetition. Hence, it passes the horizontal line test, confirming it is one-to-one.

An injective function, often synonymous with a one-to-one function, is defined by the unique mapping of elements. This means that distinct elements in the domain map to distinct elements in the co-domain. In simpler terms, no two input values share the same output value.
To check for injectivity algebraically, we assume \( f(a) = f(b) \). If this assumption leads us to \( a = b \), it confirms that the function is one-to-one. With our function, \( f(x) = -2x + 4 \), substituting gives \(-2a + 4 = -2b + 4\). Simplifying, we find \( a = b \), which confirms our function is injective. Thus, injective functions ensure every input has a unique output, making them crucial in mathematical mappings that need to be reversible or invertible.