Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = $$x^2. $$Find each of the following, if possible.(f○g)(−6)(f ○ g)(-6)

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 116

Chapter 3, Problem 116

Let $ƒ (x) = √ (x - 2)$ and $g (x) = x^{2}$ . Find each of the following, if possible.
$(f ○ g) (- 6)$

Verified step by step guidance

Understand that the notation $(f \circ g)(x)$ means the composition of functions, which is $f(g(x))$. This means you first apply $g$ to $x$, then apply $f$ to the result.

Start by finding $g(-6)$. Since $g(x) = x^2$, substitute $-6$ into $g(x)$ to get $g(-6) = (-6)^2$.

Calculate $g(-6)$ to find the value that will be input into $f$. (Note: Do not compute the final number here, just set up the expression.)

Next, substitute $g(-6)$ into $f(x) = \sqrt{x - 2}$, so you will evaluate $f(g(-6)) = \sqrt{g(-6) - 2}$.

Check the domain of $f$ to ensure the expression inside the square root is non-negative, i.e., $g(-6) - 2 \geq 0$. If this condition holds, you can proceed to simplify the expression.

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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 applying one function to the result of another, denoted as (f ○ g)(x) = f(g(x)). To evaluate (f ○ g)(-6), first find g(-6), then substitute that value into f. This process combines two functions into a single operation.

Domain of a Function

The domain is the set of all input values for which a function is defined. For f(x) = √(x-2), the expression inside the square root must be non-negative, so x-2 ≥ 0. Checking the domain ensures the function outputs real values and helps determine if the composition is possible.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function's formula and simplifying. For example, to find g(-6), replace x with -6 in g(x) = x², then calculate the result. Accurate evaluation is essential for correctly computing compositions.

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