O'Quinn Summers
03/20/2023 · Middle School
Let \( h(x)=\frac{x-2}{3 x} \) Part A Fill in the blanks to complete the rule for hoh. hoh \( (x)=\frac{-5 i x-2}{3 x-6!} \)
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Tutor-Verified Answer
Step-by-step Solution
To find \( h(h(x)) \), we first need to substitute \( h(x) \) into itself.
Given:
\[
h(x) = \frac{x-2}{3x}
\]
We want to find \( h(h(x)) \):
\[
h(h(x)) = h\left(\frac{x-2}{3x}\right)
\]
Now, we substitute \( \frac{x-2}{3x} \) into the function \( h \):
\[
h\left(\frac{x-2}{3x}\right) = \frac{\frac{x-2}{3x} - 2}{3\left(\frac{x-2}{3x}\right)}
\]
Let's simplify the numerator:
\[
\frac{x-2}{3x} - 2 = \frac{x-2 - 6x}{3x} = \frac{-5x - 2}{3x}
\]
Now, simplify the denominator:
\[
3\left(\frac{x-2}{3x}\right) = \frac{3(x-2)}{3x} = \frac{x-2}{x}
\]
Now we can write \( h(h(x)) \):
\[
h(h(x)) = \frac{\frac{-5x - 2}{3x}}{\frac{x-2}{x}} = \frac{-5x - 2}{3x} \cdot \frac{x}{x-2}
\]
This simplifies to:
\[
h(h(x)) = \frac{-5x - 2}{3(x-2)}
\]
Thus, the completed rule for \( h(h(x)) \) is:
\[
h(h(x)) = \frac{-5x - 2}{3(x-2)}
\]
Now, filling in the blanks:
hoh \( (x) = \frac{-5x - 2}{3x - 6} \) (since \( 3(x-2) = 3x - 6 \)).
Quick Answer
hoh \( (x) = \frac{-5x - 2}{3x - 6} \)
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