Warren Smith
05/26/2023 · Elementary School
\( \begin{aligned} f(x)= & 2 x^{2}-3 x+7 \quad g(x)=-4 x^{3} \quad h(x)=x^{2}-3 \\ \text { 1. } & (f+h)(x) \\ \text { 2. } & (f-h)(x)-g(x) \\ \text { 3. } & (h \cdot g)(x)+f(x) \\ \text { 4. } & \left(\frac{f}{g}\right)(x)-h(x) \\ \text { 5. } & (f(g(x))) \\ \text { 6. } & (h(g(f(x))))\end{aligned} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Let's compute each of the expressions step by step.
1. **Calculate \( (f+h)(x) \)**:
\[
f(x) + h(x) = (2x^2 - 3x + 7) + (x^2 - 3) = 2x^2 + x^2 - 3x + 7 - 3 = 3x^2 - 3x + 4
\]
So, \( (f+h)(x) = 3x^2 - 3x + 4 \).
2. **Calculate \( (f-h)(x) - g(x) \)**:
\[
f(x) - h(x) = (2x^2 - 3x + 7) - (x^2 - 3) = 2x^2 - 3x + 7 - x^2 + 3 = x^2 - 3x + 10
\]
Now subtract \( g(x) \):
\[
(f-h)(x) - g(x) = (x^2 - 3x + 10) - (-4x^3) = x^2 - 3x + 10 + 4x^3 = 4x^3 + x^2 - 3x + 10
\]
So, \( (f-h)(x) - g(x) = 4x^3 + x^2 - 3x + 10 \).
3. **Calculate \( (h \cdot g)(x) + f(x) \)**:
\[
h(x) \cdot g(x) = (x^2 - 3)(-4x^3) = -4x^5 + 12x^3
\]
Now add \( f(x) \):
\[
(h \cdot g)(x) + f(x) = (-4x^5 + 12x^3) + (2x^2 - 3x + 7) = -4x^5 + 12x^3 + 2x^2 - 3x + 7
\]
So, \( (h \cdot g)(x) + f(x) = -4x^5 + 12x^3 + 2x^2 - 3x + 7 \).
4. **Calculate \( \left(\frac{f}{g}\right)(x) - h(x) \)**:
\[
\frac{f(x)}{g(x)} = \frac{2x^2 - 3x + 7}{-4x^3}
\]
This can be simplified as:
\[
\left(\frac{f}{g}\right)(x) = -\frac{1}{4} \left( \frac{2x^2 - 3x + 7}{x^3} \right)
\]
Now subtract \( h(x) \):
\[
\left(\frac{f}{g}\right)(x) - h(x) = -\frac{1}{4} \left( \frac{2x^2 - 3x + 7}{x^3} \right) - (x^2 - 3)
\]
This expression is complex and can be left in this form, or you can combine it further if needed.
5. **Calculate \( f(g(x)) \)**:
\[
g(x) = -4x^3
\]
Now substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(-4x^3) = 2(-4x^3)^2 - 3(-4x^3) + 7 = 2(16x^6) + 12x^3 + 7 = 32x^6 + 12x^3 + 7
\]
So, \( f(g(x)) = 32x^6 + 12x^3 + 7 \).
6. **Calculate \( h(g(f(x))) \)**:
First, calculate \( f(x) \):
\[
f(x) = 2x^2 - 3x + 7
\]
Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(2x^2 - 3x + 7) = -4(2x^2 - 3x + 7)^3
\]
This is a complex expression, and we can denote it as
Quick Answer
1. \( (f+h)(x) = 3x^2 - 3x + 4 \)
2. \( (f-h)(x) - g(x) = 4x^3 + x^2 - 3x + 10 \)
3. \( (h \cdot g)(x) + f(x) = -4x^5 + 12x^3 + 2x^2 - 3x + 7 \)
4. \( \left(\frac{f}{g}\right)(x) - h(x) = -\frac{1}{4} \left( \frac{2x^2 - 3x + 7}{x^3} \right) - (x^2 - 3) \)
5. \( f(g(x)) = 32x^6 + 12x^3 + 7 \)
6. \( h(g(f(x))) = -4(2x^2 - 3x + 7)^3 \)
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