Question

if $g(x) = x^2$ and $f(x) = x + 3$
calculate $f(g(x))$

1. identify the outer and inner functions

$f(x) = x + 3$ 2. write the outer function
$f(g(x)) = x^2 + 3$ 3. substitute each $x$ with the inner function

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:

Problem 1:

Step1: Identify outer and inner functions

Outer function: \( f(x) = 2x - 3 \), Inner function: \( g(x) = 4x + 1 \)

Step2: Substitute \( g(x) \) into \( f(x) \)

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

Step3: Simplify the expression

\( 2(4x + 1) - 3 = 8x + 2 - 3 = 8x - 1 \)

Step1: Recall \( (f \circ g)(x) = f(g(x)) \)

Outer function: \( f(x) = x + 5 \), Inner function: \( g(x) = 3x - 2 \)

Step2: Substitute \( g(x) \) into \( f(x) \)

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

Step3: Simplify the expression

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

Step1: Identify outer and inner functions

Outer function: \( g(x) = 4x + 1 \), Inner function: \( f(x) = 2x - 3 \)

Step2: Substitute \( f(x) \) into \( g(x) \)

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

Step3: Simplify the expression

\( 4(2x - 3) + 1 = 8x - 12 + 1 = 8x - 11 \)

Answer:

\( 8x - 1 \)

Problem 2: