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For [tex]f(x) = x - 3[/tex] and [tex]g(x) = 5x^2 - 3[/tex], find the following functions:

a. [tex](f \circ g)(x)[/tex]

b. [tex](g \circ f)(x)[/tex]

c. [tex](f \circ g)(1)[/tex]

d. [tex](g \circ f)(1)[/tex]

Sagot :

GinnyAnsweridn
Let's solve the given problem step by step.

### a. Finding [tex]\((f \circ g)(x)\)[/tex]

The function [tex]\((f \circ g)(x)\)[/tex] means [tex]\(f(g(x))\)[/tex]. To find it, we need to plug [tex]\(g(x)\)[/tex] into the function [tex]\(f(x)\)[/tex].

Recall that:
- [tex]\(f(x) = x - 3\)[/tex]
- [tex]\(g(x) = 5x^2 - 3\)[/tex]

So, we start with [tex]\(g(x)\)[/tex]:
[tex]\[g(x) = 5x^2 - 3\][/tex]

Now, substituting [tex]\(g(x)\)[/tex] into [tex]\(f\)[/tex]:
[tex]\[f(g(x)) = f(5x^2 - 3) = (5x^2 - 3) - 3 = 5x^2 - 6\][/tex]

Hence:
[tex]\[(f \circ g)(x) = 5x^2 - 6\][/tex]

### b. Finding [tex]\((g \circ f)(x)\)[/tex]

The function [tex]\((g \circ f)(x)\)[/tex] means [tex]\(g(f(x))\)[/tex]. To find it, we need to plug [tex]\(f(x)\)[/tex] into the function [tex]\(g(x)\)[/tex].

Recall that:
- [tex]\(f(x) = x - 3\)[/tex]
- [tex]\(g(x) = 5x^2 - 3\)[/tex]

So, we start with [tex]\(f(x)\)[/tex]:
[tex]\[f(x) = x - 3\][/tex]

Now, substituting [tex]\(f(x)\)[/tex] into [tex]\(g\)[/tex]:
[tex]\[g(f(x)) = g(x - 3) = 5(x - 3)^2 - 3\][/tex]

We need to expand [tex]\(5(x - 3)^2 - 3\)[/tex]:
[tex]\[(x - 3)^2 = x^2 - 6x + 9\][/tex]
Therefore:
[tex]\[g(f(x)) = 5(x^2 - 6x + 9) - 3 = 5x^2 - 30x + 45 - 3 = 5x^2 - 30x + 42\][/tex]

Hence:
[tex]\[(g \circ f)(x) = 5x^2 - 30x + 42\][/tex]

### c. Finding [tex]\((f \circ g)(1)\)[/tex]

To find [tex]\((f \circ g)(1)\)[/tex], we need to substitute [tex]\(x = 1\)[/tex] into the function [tex]\((f \circ g)(x)\)[/tex].
Recall:
[tex]\[(f \circ g)(x) = 5x^2 - 6\][/tex]

So:
[tex]\[(f \circ g)(1) = 5(1)^2 - 6 = 5 - 6 = -1\][/tex]

Hence:
[tex]\[(f \circ g)(1) = -1\][/tex]

### d. Finding [tex]\((g \circ f)(1)\)[/tex]

To find [tex]\((g \circ f)(1)\)[/tex], we need to substitute [tex]\(x = 1\)[/tex] into the function [tex]\((g \circ f)(x)\)[/tex].
Recall:
[tex]\[(g \circ f)(x) = 5x^2 - 30x + 42\][/tex]

So:
[tex]\[(g \circ f)(1) = 5(1)^2 - 30(1) + 42 = 5 - 30 + 42 = 17\][/tex]

Hence:
[tex]\[(g \circ f)(1) = 17\][/tex]

### Summary
- [tex]\((f \circ g)(x) = 5x^2 - 6\)[/tex]
- [tex]\((g \circ f)(x) = 5x^2 - 30x + 42\)[/tex]
- [tex]\((f \circ g)(1) = -1\)[/tex]
- [tex]\((g \circ f)(1) = 17\)[/tex]
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