Question

Let a(x) and b(x) be functions. Find (a\circ b)(3) - (b\circ a)(3) if $a(x) = 2x + 5$ and $b(x) = 4 - x^2$.

Answer 1

To find \((a \circ b)(3) - (b \circ a)(3)\), we first need to compute \((a \circ b)(3)\) and \((b \circ a)(3)\).

1. **Calculate \( (a \circ b)(3) \)**:
The composition \( (a \circ b)(x) \) is defined as \( a(b(x)) \). We start by finding \( b(3) \):
\[
b(x) = 4 - x^2
\]
Therefore,
\[
b(3) = 4 - 3^2 = 4 - 9 = -5
\]

Now, we need to evaluate \( a(b(3)) = a(-5) \):
\[
a(x) = 2x + 5
\]
Thus,
\[
a(-5) = 2(-5) + 5 = -10 + 5 = -5
\]

Therefore, \( (a \circ b)(3) = -5 \).

2. **Calculate \( (b \circ a)(3) \)**:
The composition \( (b \circ a)(x) \) is defined as \( b(a(x)) \). First, we find \( a(3) \):
\[
a(3) = 2(3) + 5 = 6 + 5 = 11
\]

Now, we need to evaluate \( b(a(3)) = b(11) \):
\[
b(11) = 4 - 11^2 = 4 - 121 = -117
\]

Thus, \( (b \circ a)(3) = -117 \).

3. **Combine results**:
We now have:
\[
(a \circ b)(3) = -5
\]
\[
(b \circ a)(3) = -117
\]

Now we find \( (a \circ b)(3) - (b \circ a)(3) \):
\[
(a \circ b)(3) - (b \circ a)(3) = -5 - (-117) = -5 + 117 = 112
\]

Thus, the final result is:
\[
\boxed{112}
\]