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Chapter 0: Problem 18

Find functions \(f\) and \(g\) such that \(h=g \circ f\) (Note: The answer is not unique.) \(h(x)=\left|x^{2}-2 x+3\right|\)

Short Answer

Expert verified
In this case, one possible solution for the functions \(f\) and \(g\) is: \[f(x) = x^2 - 2x + 3\] \[g(x) = |x|\] which forms \(h(x) = |x^2 - 2x + 3|\) as a composition of functions with \(h = g \circ f\).

Step by step solution

01

Identify the quadratic function inside the absolute value

We will start by identifying the quadratic function inside the absolute value: \[h(x) = |x^2 - 2x + 3|\] The quadratic function is given by: \[f(x) = x^2 - 2x + 3\] Step 2: Rewrite the absolute value as a function
02

Rewrite the absolute value as a function

The function \(h(x)\) now can be written as the composition of these two functions: \[h(x) = |f(x)|\] Since the absolute value is a function as well, we write the function \(g(x) = |x|\). Then, we can describe \(h(x)\) as a composition of functions: \[h(x) = g(f(x))\] Step 3: Write the final answer
03

Write the functions f and g found

In this decomposition, the functions \(f\) and \(g\) are: \[f(x) = x^2 - 2x + 3\] \[g(x) = |x|\] So, the composition of functions \(h = g \circ f\) can be represented as: \[h(x) = |x^2 - 2x + 3|\]

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Most popular questions from this chapter

Write the expression in algebraic form. $$ \sin \left(\cos ^{-1} x\right) $$

Determine whether the function is one-to-one. $$ f(x)=-x^{4}+16 $$

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\sin x, \quad y=2 \sin \frac{x}{2}\)

Find the exact value of the given expression. $$ \sin ^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x^{2} \sin \frac{1}{x} $$
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