Question

if $g(x) = x^2$ and $f(x) = x + 3$
calculate $f(g(x))$
$f(x) = x + 3$
$f(g(x)) = x^2 + 3$
practice - directions: perform the indicated operation.

1. $f(x) = 2x - 3$, $g(x) = 4x + 1$

find $f(g(x))$

1. $f(x) = x + 5$, $g(x) = 3x - 2$

find $(f \circ g)(x)$

1. $f(x) = 2x - 3$, $g(x) = 4x + 1$

find $g(f(x))$

1. $f(x) = x + 5$, $g(x) = 3x - 2$

find $(g \circ f)(x)$

1. $f(x) = 5x - 1$, $g(x) = 2x + 6$

find $f(g(x))$

1. $f(x) = 3x + 2$, $g(x) = x - 4$

find $(f \circ g)(x)$

1. $f(x) = 5x - 1$, $g(x) = 2x + 6$

find $g(f(x))$

1. $f(x) = 3x + 2$, $g(x) = x - 4$

find $(g \circ f)(x)$

Explanation:

Let's solve each problem one by one:

Problem 1:

Given \( f(x) = 2x - 3 \) and \( g(x) = 4x + 1 \), find \( f(g(x)) \).

Step 1: Identify the inner function

The inner function is \( g(x) = 4x + 1 \).

Step 2: Substitute \( g(x) \) into \( f(x) \)

Replace every \( x \) in \( f(x) \) with \( g(x) \). So we have \( f(g(x)) = 2(g(x)) - 3 \).

Step 3: Substitute \( g(x) = 4x + 1 \) into the expression

\( f(g(x)) = 2(4x + 1) - 3 \)

Step 4: Simplify the expression

First, distribute the 2: \( 2(4x + 1) = 8x + 2 \)
Then, subtract 3: \( 8x + 2 - 3 = 8x - 1 \)

Step 1: Identify the inner function

The inner function is \( g(x) = 3x - 2 \).

Step 2: Substitute \( g(x) \) into \( f(x) \)

Replace every \( x \) in \( f(x) \) with \( g(x) \). So we have \( f(g(x)) = g(x) + 5 \).

Step 3: Substitute \( g(x) = 3x - 2 \) into the expression

\( f(g(x)) = (3x - 2) + 5 \)

Step 4: Simplify the expression

Combine like terms: \( 3x - 2 + 5 = 3x + 3 \)

Step 1: Identify the inner function

The inner function is \( f(x) = 2x - 3 \).

Step 2: Substitute \( f(x) \) into \( g(x) \)

Replace every \( x \) in \( g(x) \) with \( f(x) \). So we have \( g(f(x)) = 4(f(x)) + 1 \).

Step 3: Substitute \( f(x) = 2x - 3 \) into the expression

\( g(f(x)) = 4(2x - 3) + 1 \)

Step 4: Simplify the expression

First, distribute the 4: \( 4(2x - 3) = 8x - 12 \)
Then, add 1: \( 8x - 12 + 1 = 8x - 11 \)

Answer:

\( f(g(x)) = 8x - 1 \)

Problem 2:

Given \( f(x) = x + 5 \) and \( g(x) = 3x - 2 \), find \( (f \circ g)(x) \) (which is the same as \( f(g(x)) \)).