Question

A function \(a\) satisfies \(a(-2)=3\) and \(a(3)=8 .\) A function \(b\) satisfies \(b(3)=4\) and \(b(7)=-2 .\) What is the value of \(a(b(7)) ?\)

Short answer

a(b(7)) = 3

Step-by-step solution

Identify the value of b(7)

According to the problem, the function b satisfies the condition that when the input is 7, the output is -2. Therefore, we have: b(7) = -2

Substitute b(7) into a(x)

Next, we need to find the value of the function a when the input is -2 (since b(7) = -2). According to the problem, the function a satisfies the condition that when the input is -2, the output is 3. Therefore, we have: a(-2) = 3

Combine the values to find a(b(7))

Now, combining the results of the previous steps, we get: a(b(7)) = a(-2) = 3

Key concepts

Function Composition

Function composition involves creating a new function by applying one function to the results of another.
In mathematical notation, if we have two functions, say function a and function b, the composition of a with b is written as \(a(b(x))\).
This means you first apply function b to the input x, and then apply function a to the result of \(b(x)\).
This is a crucial concept in mathematics because it allows you to link two processes or calculations into one step.
In this exercise, we are asked to evaluate \(a(b(7))\).

Problem-Solving Steps

To solve problems involving function composition, follow these key steps:

• First, understand the functions and the given conditions.
• Identify the immediate function and evaluate it with the given input.
• Next, take this result and substitute it into the next function.
• Combine the two results to find the final answer.

In our exercise:
Step 1: Evaluate \(b(7)\) and find that \(b(7) = -2\).
Step 2: Substitute \(b(7)\) into function a to get \(a(-2)\).
Step 3: Since \(a(-2) = 3\), the final result is \(a(b(7)) = 3\).

SAT Math Problems

Function composition is a common topic in SAT math problems.
These questions test your ability to follow through with multiple steps and understand how to apply different functions together.
When facing these questions:

• Carefully read the problem and extract critical information.
• Note the functions given and their respective conditions.
• Perform the operations step by step, making sure not to skip any details.

In many cases, breaking the problem down into smaller steps and solving each step methodically is key to success.

Input-Output Relationship

Functions describe a specific relationship between input (x) and output (y).
This means for every x-value you put into a function, you get a specific y-value out.
Understanding this relationship is crucial when dealing with function composition.
For example:

• If \(b(7) = -2\), then -2 is the output when 7 is the input to function b.
• If \(a(-2) = 3\), then 3 is the output when -2 is the input to function a.

In function composition, you use the output of one function as the input to another.
This helps in chaining operations together to solve more complex problems in a structured way.