For the functions $$ f(x)=2 x^{2}-3 x-8 $$, and $$ g(x)=x-3 $$, find $$ (f \circ g)(x) $$ a) $$ 2 x^{2}-15 x+19 $$ b) $$ 2 x^{2}-12 x+9 $$ c) $$ 2 x^{2}-12+19 $$ d) $$ 2 x^{2}-6 x+10 $$ e) $$ 2 x^{2}-3 x-11 $$

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UpStudy AI · Accepted Answer

We are given:
$$
f(x)=2x^2-3x-8,\quad g(x)=x-3.
$$

To find $$(f \circ g)(x)$$, we substitute $$g(x)$$ into $$f(x)$$:
$$
(f \circ g)(x)=f(g(x))=2(g(x))^2-3(g(x))-8.
$$

Since $$g(x)=x-3$$, we have:
$$
(f \circ g)(x)=2(x-3)^2-3(x-3)-8.
$$

**Step 1: Expand $$(x-3)^2$$**
$$
(x-3)^2 = x^2 - 6x + 9.
$$

**Step 2: Substitute back and expand**
$$
2(x-3)^2 = 2(x^2-6x+9)=2x^2-12x+18.
$$
$$
-3(x-3) = -3x+9.
$$

Thus,
$$
(f \circ g)(x)=2x^2-12x+18-3x+9-8.
$$

**Step 3: Combine like terms**
- $$x^2$$ terms: $$2x^2$$.
- $$x$$ terms: $$-12x-3x=-15x$$.
- Constant terms: $$18+9-8=19$$.

So, we get:
$$
(f \circ g)(x)=2x^2-15x+19.
$$

The correct answer is option (a).
$$ (f \circ g)(x) = 2x^2 - 15x + 19 $$, so the answer is option (a).

For the functions \( f(x)=2 x^{2}-3 x-8 \), and \( g(x)=x-3 \), find \( (f \circ g)(x) \) a) \( 2 x^{2}-15 x+19 \) b) \( 2 x^{2}-12 x+9 \) c) \( 2 x^{2}-12+19 \) d) \( 2 x^{2}-6 x+10 \) e) \( 2 x^{2}-3 x-11 \)

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