Answer:
f(x) = x² + 5
Explanation:
In order to write the equation y - x² = 5 in function notation, we first need to express y as a function of x.
The given equation represents a relationship between x and y, where the value of y is determined by the value of x. To write this as a function, we can rearrange the equation to solve for y:
⇒ y - x² = 5
⇒ y - x² + x² = 5 + x²
⇒ y = 5 + x²
∴ y = x² + 5
Now, we can represent this relationship as a function, which we'll call "f," by writing:
⇒ f(x) = x² + 5
In this notation, f(x) represents the output (or value of y) that the function produces when the input is x. The function f takes x as an input, squares it, and then adds 5 to the result to give us the value of y.
Terminology and concepts:
Function notation: Function notation is a way of representing a relationship between an input (usually denoted as x) and an output (usually denoted as y) using a specific rule or formula. In function notation, f(x) is read as "f of x" and represents the value of the function, "f," when the input is x. We are not limited to using just the letter "f." We can use other letters such as g or h (f(x), g(x), h(x), ...). As you level up in math, you'll find that functions can depend on more than just one variable. Such as x and y or x, y, and z (f(x, y) or f(x, y, z)).
Dependent and independent variables: In the context of functions, y or f(x) is the dependent variable because its value depends on the value of the independent variable, x. The independent variable is the input to the function, and the dependent variable is the output that the function produces based on that input.
Solving for a variable: In the given context, "solving for y" means isolating y on one side of the equation, so that the equation is in the form y = something. This allows us to express y explicitly as a function of x.
Equation rearrangement: Manipulating the equation to change its form without altering the relationship it represents. In this case, we rearranged the equation y - x² = 5 to solve for y.