If ƒ(x) = √x and g(x) = x³-2 and , simplify the expressions (ƒ o g) (3), (ƒ o ƒ) (64), (g o ƒ) (x) and (ƒ o g) (x)

Verified step by step guidance

Step 1: Understand the composition of functions. The notation (f o g)(x) means f(g(x)), which means you first apply g to x, and then apply f to the result.

Step 2: Calculate (f o g)(3). First, find g(3) by substituting 3 into g(x) = x^3 - 2, then apply f to the result.

Step 3: Calculate (f o f)(64). First, find f(64) by substituting 64 into f(x) = $\sqrt{x}$, then apply f to the result.

Step 4: Calculate (g o f)(x). First, find f(x) by substituting x into f(x) = $\sqrt{x}$, then apply g to the result.

Step 5: Calculate (f o g)(x). First, find g(x) by substituting x into g(x) = x^3 - 2, then apply f to the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. For example, if you have functions f(x) and g(x), the composition (f o g)(x) means you first apply g to x, then apply f to the result of g. Understanding this concept is crucial for simplifying expressions involving multiple functions.

Square Root Function

The square root function, denoted as f(x) = √x, is defined for non-negative values of x and returns the principal square root. This function is essential in the given problem as it affects the output of the composed functions, particularly when evaluating expressions like (f o g)(3) and (f o f)(64). Recognizing the domain restrictions of the square root function is important for valid outputs.

Polynomial Functions

Polynomial functions, such as g(x) = x³ - 2, are expressions that involve variables raised to whole number powers. They are continuous and differentiable everywhere on their domain. In the context of the question, understanding how to evaluate and manipulate polynomial functions is necessary for simplifying expressions like (g o f)(x) and (f o g)(x).

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Related Practice

Textbook Question

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>

Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.

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Textbook Question

Use the graph of $f$ in the figure to plot the following functions.

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$y=f$\left$(x-2$\right$)$