https://file.aiquickdraw.com/mathbot/file/1726904835389zj0tvv79.jpg

The image contains the following mathematical functions and tasks:

### Functions:
1. \( a(x) = -3x + 5 \)
2. \( b(x) = 11x - 1 \)
3. \( c(x) = -8x - 3 \)

### Tasks:
The tasks require evaluating specific expressions involving these functions.

#### Problems:
1. \( (a - c)^{-1}(1) \)
2. \( (a + c)^{-1}(0) \)
3. \( (a - (c + b)^2)^{-1}(-1) \)

These involve subtracting or adding the functions \( a(x) \), \( b(x) \), and \( c(x) \), and then finding the inverse of the resulting expression evaluated at a specific value.

### Let's break down the tasks:

1. **For \( (a - c)^{-1}(1) \)**:
   - First, compute \( a(x) - c(x) \).
   - Then, find the inverse of the resulting function.
   - Finally, evaluate the inverse at \( x = 1 \).

2. **For \( (a + c)^{-1}(0) \)**:
   - First, compute \( a(x) + c(x) \).
   - Then, find the inverse of the resulting function.
   - Finally, evaluate the inverse at \( x = 0 \).

3. **For \( (a - (c + b)^2)^{-1}(-1) \)**:
   - First, compute \( c(x) + b(x) \), square the result.
   - Subtract this from \( a(x) \).
   - Then, find the inverse of the resulting function.
   - Finally, evaluate the inverse at \( x = -1 \).

Would you like me to walk you through the steps to solve each of these problems in detail?

### Related Questions:
1. What does it mean for a function to have an inverse?
2. How do you compute the inverse of a linear function?
3. What operations are necessary to simplify expressions like \( a(x) - c(x) \)?
4. How do you handle squared terms like \( (c + b)^2 \) when combining functions?
5. How do we evaluate a function's inverse at a particular value?

### Tip:
When finding the inverse of a linear function, remember that the inverse function essentially "undoes" the original operation by switching \( x \) and \( y \), then solving for \( y \).

Solve the given expressions involving the functions a(x), b(x), and c(x). The tasks include evaluating inverses of expressions formed by the functions.

This math problem involves evaluating inverse functions for given linear equations a(x), b(x), and c(x). Students will learn how to calculate and analyze these inverse functions step-by-step. 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