The functions f and g for x∈ 𝑅 by f(x) = x – 2 and g(x) = ax + b, where a , b ∈ 𝑅. Given that
f ° g(2) = -3 and g ° 𝑓 (1) = 5, find the value of a and the value of b.

We are given the following functions:

- \( f(x) = x - 2 \)
- \( g(x) = ax + b \)

Also, we know that:

1. \( f \circ g(2) = -3 \)
2. \( g \circ f(1) = 5 \)

We need to find the values of \( a \) and \( b \). Let's solve this step by step.

### Step 1: Use \( f \circ g(2) = -3 \)

This means we first apply \( g \) to 2, and then apply \( f \) to the result of \( g(2) \):

\[
f(g(2)) = -3
\]

Now, substitute \( g(x) = ax + b \):

\[
g(2) = a(2) + b = 2a + b
\]

Next, apply \( f \) to \( g(2) \):

\[
f(2a + b) = (2a + b) - 2
\]

We are told that \( f(g(2)) = -3 \), so:

\[
(2a + b) - 2 = -3
\]

Solving this equation:

\[
2a + b - 2 = -3
\]
\[
2a + b = -1 \quad \text{(Equation 1)}
\]

### Step 2: Use \( g \circ f(1) = 5 \)

This means we first apply \( f \) to 1, and then apply \( g \) to the result of \( f(1) \):

\[
g(f(1)) = 5
\]

Now, find \( f(1) \):

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

Next, apply \( g \) to \( f(1) = -1 \):

\[
g(-1) = a(-1) + b = -a + b
\]

We are told that \( g(f(1)) = 5 \), so:

\[
-a + b = 5 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations

We now have the following system of equations:

1. \( 2a + b = -1 \)
2. \( -a + b = 5 \)

Let's solve this system.

From Equation 2, solve for \( b \):

\[
b = a + 5
\]

Substitute this into Equation 1:

\[
2a + (a + 5) = -1
\]
\[
3a + 5 = -1
\]
\[
3a = -6
\]
\[
a = -2
\]

Now substitute \( a = -2 \) into the equation \( b = a + 5 \):

\[
b = -2 + 5 = 3
\]

### Final Answer:

- \( a = -2 \)
- \( b = 3 \)

Let me know if you need more details or clarifications!

---

Here are some related questions for further exploration:

1. What is the composite function \( f \circ g(x) \)?
2. How do you solve systems of linear equations algebraically?
3. What is the importance of composition in function theory?
4. Can the values of \( a \) and \( b \) be interpreted geometrically?
5. How would you approach a similar problem where more compositions are involved?

**Tip:** When solving a system of equations, always try to express one variable in terms of the other to simplify the process.

The functions f and g for x∈ 𝑅 by f(x) = x – 2 and g(x) = ax + b, where a , b ∈ 𝑅. Given that f ° g(2) = -3 and g ° f(1) = 5, find the value of a and the value of b.

Learn how to find the values of a and b for the functions f(x) = x - 2 and g(x) = ax + b given the conditions f ° g(2) = -3 and g ° f(1) = 5. This detailed solution walks through function composition and solving systems of linear equations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation