f(x)=1/x ; h(x)=x^4-2x^2+1            find (h•f)(x)

To find \( (h \circ f)(x) \), we need to compose the functions \( h(x) \) and \( f(x) \). This means we will substitute \( f(x) \) into \( h(x) \).

Given:
\[
f(x) = \frac{1}{x}
\]
and
\[
h(x) = x^4 - 2x^2 + 1
\]

The composition \( (h \circ f)(x) = h(f(x)) \) is found by substituting \( f(x) \) into \( h(x) \):

\[
(h \circ f)(x) = h\left( \frac{1}{x} \right)
\]

Now substitute \( \frac{1}{x} \) into \( h(x) \):

\[
h\left( \frac{1}{x} \right) = \left( \frac{1}{x} \right)^4 - 2\left( \frac{1}{x} \right)^2 + 1
\]

Simplify each term:

\[
\left( \frac{1}{x} \right)^4 = \frac{1}{x^4}, \quad \left( \frac{1}{x} \right)^2 = \frac{1}{x^2}
\]

Substitute these into the equation:

\[
(h \circ f)(x) = \frac{1}{x^4} - 2 \cdot \frac{1}{x^2} + 1
\]

Thus, the composition is:

\[
(h \circ f)(x) = \frac{1}{x^4} - \frac{2}{x^2} + 1
\]

This is the simplified form of \( (h \circ f)(x) \).

Would you like more details on function composition or any further explanation?

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Here are 5 related questions for further practice:
1. Find \( (f \circ h)(x) \) for the given functions.
2. What is the domain of \( (h \circ f)(x) \)?
3. If \( f(x) = \frac{1}{x} \) and \( g(x) = x^2 + 3x + 2 \), find \( (g \circ f)(x) \).
4. Solve \( (h \circ f)(x) = 0 \) for \( x \).
5. What is the derivative of \( (h \circ f)(x) \)?

**Tip:** When composing functions, always pay attention to the domain of the resulting function to ensure the composition is well-defined.

f(x)=1/x ; h(x)=x^4-2x^2+1 find (h•f)(x)

Learn how to find the composition of functions (h • f)(x) where f(x) = 1/x and h(x) = x^4 - 2x^2 + 1. This step-by-step guide covers key mathematical concepts including function composition, rational functions, and polynomials, suitable for high school students in Grades 10-12.

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