State the Implied Domain and Range of each of the following functions. If f −1 (x) exists, determine f −1 (x) (a) f(x) = − p (x − 3)2. (b) f(x) = −e (x+2)2 . (c) f(x) = ln(3 − 2x) + 7.

Let's analyze each function to determine their implied domain, range, and whether their inverse function \( f^{-1}(x) \) exists.

### (a) \( f(x) = -\sqrt{(x - 3)^2} \)

#### Domain:
- The expression under the square root, \( (x - 3)^2 \), is non-negative for all \( x \) since it's a square.
- Therefore, the function \( f(x) \) is defined for all real values of \( x \).

**Domain:** \( (-\infty, \infty) \)

#### Range:
- \( f(x) = -\sqrt{(x - 3)^2} \) simplifies to \( f(x) = -(x - 3) \) (since \( \sqrt{(x - 3)^2} \) is always non-negative and represents the absolute value).
- The function simplifies to \( f(x) = -(x - 3) \), which is a linear function with a negative slope.
- Since it’s a linear function with a negative slope, it decreases without bound as \( x \) increases.

**Range:** \( (-\infty, \infty) \)

#### Inverse:
- Given \( y = -(x - 3) \), solve for \( x \) in terms of \( y \):
\[
y = -x + 3
\]
\[
x = 3 - y
\]
- Therefore, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = 3 - x
\]

**Inverse:** \( f^{-1}(x) = 3 - x \)

### (b) \( f(x) = -e^{(x+2)^2} \)

#### Domain:
- The function \( e^{(x+2)^2} \) is defined for all real numbers \( x \) because the exponent is a real number.

**Domain:** \( (-\infty, \infty) \)

#### Range:
- Since \( e^{(x+2)^2} \) is always positive and strictly increasing, the function \( -e^{(x+2)^2} \) will always be negative.
- The minimum value (closest to 0) occurs when \( x = -2 \), and the function approaches \( -\infty \) as \( x \) moves away from \(-2\).

**Range:** \( (-\infty, -1] \)

#### Inverse:
- The function \( f(x) = -e^{(x+2)^2} \) is not one-to-one because it is not injective; it does not have a unique \( y \) value for each \( x \).
- Therefore, the inverse \( f^{-1}(x) \) does not exist.

**Inverse:** Does not exist.

### (c) \( f(x) = \ln(3 - 2x) + 7 \)

#### Domain:
- The logarithm function \( \ln(3 - 2x) \) is defined when \( 3 - 2x > 0 \).
- Solve the inequality:
\[
3 - 2x > 0
\]
\[
x < \frac{3}{2}
\]
- Thus, \( f(x) \) is defined for \( x < \frac{3}{2} \).

**Domain:** \( (-\infty, \frac{3}{2}) \)

#### Range:
- Since \( \ln(3 - 2x) \) can take all real values as \( x \) approaches its domain limits, the entire real number line is covered when you add 7.

**Range:** \( (-\infty, \infty) \)

#### Inverse:
- To find the inverse, set \( y = \ln(3 - 2x) + 7 \) and solve for \( x \):
\[
y - 7 = \ln(3 - 2x)
\]
\[
e^{y - 7} = 3 - 2x
\]
\[
2x = 3 - e^{y - 7}
\]
\[
x = \frac{3 - e^{y - 7}}{2}
\]
- Therefore, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = \frac{3 - e^{x - 7}}{2}
\]

**Inverse:** \( f^{-1}(x) = \frac{3 - e^{x - 7}}{2} \)

Would you like further details or have any questions?

### Related Questions
1. What are the general properties of logarithmic functions in terms of their domain and range?
2. How does the exponential function's range affect the existence of its inverse?
3. Can we graphically determine if a function has an inverse?
4. How do transformations like shifting and scaling affect the domain and range of basic functions?
5. What methods can we use to verify if a function is one-to-one?

### Tip
Always remember to check the domain carefully, especially for functions involving square roots, logarithms, or fractions. These often have restrictions that affect the existence of the function and its inverse.

Explore the implied domain, range, and existence of inverse functions for f(x) = -sqrt((x - 3)^2), f(x) = -e^((x + 2)^2), and f(x) = ln(3 - 2x) + 7. Understand the mathematical concepts behind these functions with detailed solutions and insights.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation