Question

Inverse function top part and composition of functions bottom part

please review any mistakes i did and help me with number 3 pls!!!

Answer 1

In a mathematical context, h(g(2)) is 6, f(h(x)) is
`\(5(x^2 + 2)\)`. For given points and a table, f(g(2)) is -3/2, and g(f(5)) is 2. These computations involve function compositions and values from respective functions.

Let's break down the given expressions step by step:

1. Given functions:

- f(x) = 5x

-
`\( h(x) = x^2 + 2 \)`

- g(x) = x - 4

a. h(g(2)):

`\[ h(g(2)) = h(2 - 4) = h(-2) = (-2)^2 + 2 = 6 \]`

b. f(h(x)):

`\[ f(h(x)) = f(x^2 + 2) = 5(x^2 + 2) \]`

2. Given points for f(x) and the table for g(x):

f(x) points: (-2,2), (0,2), (3,4), (5,0), (6,-2), (8,-1)

g(x) table: x = -3, -1, 0, 2, 3, 5 and y = -5, -4, 2, 7, -1, 8

a. f(g(2)):

g(2) = 7.

Since there's no x = 7 in the graph, approximate using points (6, -2) and (8, -1):

`\((1)/(2)\) slope. \( f(7) \approx -2 + (1)/(2) \cdot (7 - 6) = -(3)/(2) \).`

Therefore,
`\( f(g(2)) \approx -(3)/(2) \).`

b. g(f(5)):

Find f(5) first using the coordinates for f(x) where x = 5 (the point (5,0)):

`\[ f(5) = 5 * 0 = 0 \]`

Now, find g(f(5)):

g(f(5)) = g(0)

Use the table for g(x) where x = 0 (the point (0,2)):

g(0) = 2

Therefore, the answers are:

1. a. h(g(2)) = 6

b.
`\( f(h(x)) = 5(x^2 + 2) \)`

2. a. f(g(2)) = -3/2

b. g(f(5)) = 2