evaluate the function as indicated. Determine its domain and range. 29.f􏰀x􏰁􏰡􏰄2x􏰢1, x<0 (a) f􏰀􏰣1􏰁 (b) f􏰀0􏰁
(c) f􏰀2􏰁
(d) f􏰀t2 􏰢 1􏰁

Let's break down the problem. The function provided is:

\[
f(x) = 2x + 1, \text{ for } x < 0
\]

The function is only defined for \(x < 0\), meaning we can only evaluate \(f(x)\) for values less than zero.

Now, let's evaluate each part:

### (a) \( f(-1) \)
We need to find \(f(-1)\). Since \(-1 < 0\), we can apply the function rule:

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

So, \( f(-1) = -1 \).

### (b) \( f(0) \)
The function is only defined for \(x < 0\). Since \(0 \not< 0\), \(f(0)\) is **undefined**.

### (c) \( f(2) \)
Similarly, \(2 \not< 0\), so \(f(2)\) is **undefined** as well.

### (d) \( f(t^2 + 1) \)
Here, we are asked to evaluate \(f(t^2 + 1)\). Since \(t^2 + 1\) is always positive (for all real values of \(t\)), and the function is only defined for \(x < 0\), \(f(t^2 + 1)\) is also **undefined**.

---

### Domain
The domain of \(f(x)\) is the set of \(x\)-values for which the function is defined. In this case, the function is only defined for \(x < 0\), so the domain is:

\[
\text{Domain: } (-\infty, 0)
\]

### Range
To find the range, we analyze the output values of \(f(x) = 2x + 1\) for \(x < 0\). As \(x\) approaches \(0\) from the left (i.e., \(x \to 0^-\)), \(f(x)\) approaches \(1\). As \(x\) becomes more negative (i.e., \(x \to -\infty\)), \(f(x)\) decreases without bound. Thus, the range is:

\[
\text{Range: } (-\infty, 1)
\]

### Summary of Results:
- \(f(-1) = -1\)
- \(f(0)\) is undefined
- \(f(2)\) is undefined
- \(f(t^2 + 1)\) is undefined

Domain: \((- \infty, 0)\)

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

---

Do you need more details or have any questions?

Here are 5 related questions:
1. How would the domain change if the function were defined for \(x \geq 0\)?
2. What happens to the range if we modify the function to \(f(x) = 2x - 1\)?
3. How can we check if a function is continuous at a specific point?
4. What is the interpretation of a function being undefined for certain values of \(x\)?
5. How does the slope of the function \(f(x) = 2x + 1\) affect its graph?

**Tip:** The domain of a function is the set of input values that don't violate any conditions set by the function's definition (e.g., restrictions on \(x\) like \(x < 0\) or square roots requiring non-negative arguments).

Evaluate the function as indicated. Determine its domain and range. 29. f(x) = 2x + 1, x < 0 (a) f(-1) (b) f(0) (c) f(2) (d) f(t^2 + 1)

Learn how to evaluate the function f(x) = 2x + 1 for various inputs and determine its domain and range. This problem involves understanding functions, their restrictions, and how to find the domain and range. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation