Composite functions and notation
Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
F(y⁴)

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Identify the function F(x) given as F(x) = $\frac{1}{x-3}$.

Substitute y^4 for x in the function F(x) to find F(y^4).

This substitution gives F(y^4) = $\frac{1}{y^4 - 3}$.

Simplify the expression if possible, but in this case, it is already in its simplest form.

Conclude that F(y^4) = $\frac{1}{y^4 - 3}$ is the simplified expression.

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Composite Functions

Composite functions are formed when one function is applied to the result of another function. In this case, if we have functions f and g, the composite function (f ∘ g)(x) means f(g(x)). Understanding how to evaluate composite functions is crucial for simplifying expressions like F(y⁴), where F is applied to the output of another function.

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Function Notation

Function notation is a way to denote functions and their inputs clearly. For example, F(x) indicates that F is a function of x. In the expression F(y⁴), y⁴ is the input to the function F. Familiarity with function notation helps in correctly interpreting and manipulating expressions involving multiple functions.

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Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with. This can include combining like terms, factoring, or substituting values. In the context of the given question, simplifying F(y⁴) requires substituting y⁴ into the function F and then performing any necessary algebraic operations to express the result in a more manageable form.

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Pratica correlata

Domanda del libro di testo

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m² is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t$\geq{0}$$ is $d$\left$(t$\right$)=$\left$(5-0.22t$\right$)^2$ .