Problem 19 Find functions $f$ and $g$ s... [FREE SOLUTION]

Chapter 0: Problem 19

Find functions $f$ and $g$ such that $h=g \circ f$ (Note: The answer is not unique.) $h(x)=\frac{1}{\sqrt{x^{2}-4}}$

Short Answer

Expert verified

One possible solution is the following functions: $$f(x) = x^2 - 4$$ $$g(x) = \frac{1}{\sqrt{x}}$$ These functions satisfy the condition $h(x) = g(f(x))$, as $h(x) = g(f(x)) = g(x^2 - 4) = \frac{1}{\sqrt{x^2 - 4}}$. Note that the answer is not unique and there may be other possible pairs of functions for this composition.

Step by step solution

Identify possible ways to break down $h(x)$

We need to find two functions, $f(x)$ and $g(x)$, such that $h(x) = \frac{1}{\sqrt{x^2 - 4}}$ can be written as a composition of these two functions: $h(x)=g(f(x))$. Let's first try to break down the given function $h(x)$ by focusing on the inner part of the expression, $x^{2}-4$. We can use a simple function to represent the inner expression. Let's choose the function $f(x)=x^{2}-4$. Then, we need to find another function $g(x)$ that when applied to $f(x)$, results in $h(x)$.

Create the second function and test it

Now we need a function $g(x)$ such that when we apply it to $f(x) = x^2 - 4$, it results in $h(x) = \frac{1}{\sqrt{x^2 - 4}}$. Based on the form of $h(x)$, we can try to find a function $g(x)$ of the form $g(x) = \frac{1}{\sqrt{x}}$. We will test if this function $g(x)$ results in the original function $h(x)$ when applied to $f(x)$. Let's compute $g(f(x))$ using the functions we've chosen for $f(x)$ and $g(x)$: $$g(f(x)) = g(x^2 - 4) = \frac{1}{\sqrt{x^2 - 4}}$$ Notice that $g(f(x)) = h(x)$, so we have found two functions $f(x)$ and $g(x)$ that, when composed, result in the given function $h(x)$.

Write down the final result

We have found the following functions that satisfy the condition $h(x) = g(f(x))$: $$f(x) = x^2 - 4$$ $$g(x) = \frac{1}{\sqrt{x}}$$ So, $h(x) = g(f(x)) = g(x^2 - 4) = \frac{1}{\sqrt{x^2 - 4}}$. Remember that this is not the unique solution, and there may be other possible pairs of functions that can be the composition of $h(x)$.

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91影视

Find functions \(f\) and \(g\) such that \(h=g \circ f\) (Note: The answer is not unique.) \(h(x)=\frac{1}{\sqrt{x^{2}-4}}\)

Short Answer

Step by step solution

Identify possible ways to break down \(h(x)\)

Create the second function and test it

Write down the final result

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