Given the functions P(x) = 2x+1 and g(x) = x−5, find the composite functions gp(x), pg(x), pp(x), and gg(x).

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the functions P(x) = 2x+1 and g(x) = x−5. a) Find gp(x) Step 1: To find gp(x), we substitute P(x) into g(x). gp(x) = g(P(x)) = g(2x+1); Step 2: Substitute 2x+1 into g(x) = x−5. g(2x+1) = (2x+1) − 5 Step 3: Simplify the expression. g(2x+1) = 2x − 4 The expression for gp(x) is 2x−4. b) Find pg(x) Step 1: To find pg(x), we substitute g(x) into P(x). pg(x) = P(g(x)) = P(x−5); Step 2: Substitute x−5 into P(x) = 2x+1. P(x−5) = 2(x−5) + 1 Step 3: Expand and simplify the expression. P(x−5) = 2x − 10 + 1; P(x−5) = 2x − 9 The expression for pg(x) is 2x−9. c) Find pp(x) Step 1: To find pp(x), we substitute P(x) into P(x). pp(x) = P(P(x)) = P(2x+1); Step 2: Substitute 2x+1 into P(x) = 2x+1. P(2x+1) = 2(2x+1) + 1 Step 3: Expand and simplify the expression. P(2x+1) = 4x + 2 + 1; P(2x+1) = 4x + 3 The expression for pp(x) is 4x+3. d) Find gg(x) Step 1: To find gg(x), we substitute g(x) into g(x). gg(x) = g(g(x)) = g(x−5); Step 2: Substitute x−5 into g(x) = x−5. g(x−5) = (x−5) − 5 Step 3: Simplify the expression. g(x−5) = x − 10 The expression for gg(x) is x−10.

Given the functions P(x) = 2x+1 and g(x) = x−5, find the composite functions gp(x), pg(x), pp(x), and gg(x).

Answers

Step-by-step solution

Part a· 3 steps

To find $gp(x)$ , we substitute $P(x)$ into $g(x)$ .

Substitute $2x+1$ into $g(x) = x-5$ .

Simplify the expression.

Part b· 3 steps2x-9

To find $pg(x)$ , we substitute $g(x)$ into $P(x)$ .

Substitute $x-5$ into $P(x) = 2x+1$ .

Expand and simplify the expression.

Part c· 3 steps4x+3

To find $pp(x)$ , we substitute $P(x)$ into $P(x)$ .

Substitute $2x+1$ into $P(x) = 2x+1$ .

Expand and simplify the expression.

Part d· 3 stepsx-10

To find $gg(x)$ , we substitute $g(x)$ into $g(x)$ .

Substitute $x-5$ into $g(x) = x-5$ .

Simplify the expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Given the functions P(x) = 2x+1 and g(x) = x−5, find the composite functions gp(x), pg(x), pp(x), and gg(x).

Answers

Step-by-step solution

Part a· 3 steps

To find gp(x)gp(x)gp(x), we substitute P(x)P(x)P(x) into g(x)g(x)g(x).

Substitute 2x+12x+12x+1 into g(x)=x−5g(x) = x-5g(x)=x−5.

Simplify the expression.

Part b· 3 steps2x-9

To find pg(x)pg(x)pg(x), we substitute g(x)g(x)g(x) into P(x)P(x)P(x).

Substitute x−5x-5x−5 into P(x)=2x+1P(x) = 2x+1P(x)=2x+1.

Expand and simplify the expression.

Part c· 3 steps4x+3

To find pp(x)pp(x)pp(x), we substitute P(x)P(x)P(x) into P(x)P(x)P(x).

Substitute 2x+12x+12x+1 into P(x)=2x+1P(x) = 2x+1P(x)=2x+1.

Expand and simplify the expression.

Part d· 3 stepsx-10

To find gg(x)gg(x)gg(x), we substitute g(x)g(x)g(x) into g(x)g(x)g(x).

Substitute x−5x-5x−5 into g(x)=x−5g(x) = x-5g(x)=x−5.

Simplify the expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

To find $gp(x)$ , we substitute $P(x)$ into $g(x)$ .

Substitute $2x+1$ into $g(x) = x-5$ .

To find $pg(x)$ , we substitute $g(x)$ into $P(x)$ .

Substitute $x-5$ into $P(x) = 2x+1$ .

To find $pp(x)$ , we substitute $P(x)$ into $P(x)$ .

Substitute $2x+1$ into $P(x) = 2x+1$ .

To find $gg(x)$ , we substitute $g(x)$ into $g(x)$ .

Substitute $x-5$ into $g(x) = x-5$ .