Question

Find (fog)(4) and (gof)(4).
f(x)=2x-1; g(x)=x +3
(fog)(4) =
(gof)(4) =

Answer 1

`\[ (f \circ g)(4) = 13 \]`

`\[ (g \circ f)(4) = 10 \]`

Explanation:

Given the functions:

`\[ f(x) = 2x - 1 \]`

`\[ g(x) = x + 3 \]`

1. First, let's find
`\( f \circ g \)`:

`\[ (f \circ g)(x) = f(g(x)) \]`

2. Substitute
`\( g(x) \)` into
`\( f \)`:

`\[ (f \circ g)(x) = 2(g(x)) - 1 \]`

`\[ (f \circ g)(x) = 2(x + 3) - 1 \]`

`\[ (f \circ g)(x) = 2x + 6 - 1 \]`

`\[ (f \circ g)(x) = 2x + 5 \]`

3. To find
`\( (f \circ g)(4) \)`, substitute
`\( x = 4 \)`:

`\[ (f \circ g)(4) = 2(4) + 5 = 8 + 5 = 13 \]`

4. Now, for
`\( g \circ f \)`:

`\[ (g \circ f)(x) = g(f(x)) \]`

5. Substitute
`\( f(x) \)` into
`\( g \)`:

`\[ (g \circ f)(x) = (2x - 1) + 3 \]`

`\[ (g \circ f)(x) = 2x + 2 \]`

6. To find
`\( (g \circ f)(4) \)`, substitute
`\( x = 4 \)`:

`\[ (g \circ f)(4) = 2(4) + 2 = 8 + 2 = 10 \]`

So,

`\[ (f \circ g)(4) = 13 \]`

`\[ (g \circ f)(4) = 10 \]`