Question: For the following function, $f(x)$, find $f^{-1}(x)$ and verify that they are inverses.
f(x) = \sqrt{2x + 1}
1. According to a website, a test of the temperature of a hot liquid after time in a well-known brand thermos can be represented by the chart below. Answer the questions below based on the chart.
t (hours) 0 6 12 18 24.5 30.5
H(t) (degrees F) 212 207 190 164 149 143
a. Find $H(18)$ and interpret this value in the context of this problem.
b. Find $H^{-1}(164)$ and interpret this value in the context of this problem.
c. Find $H^{-1}(212)$ and interpret this value in the context of the problem.
2. Find the inverse of $f(x) = \sqrt{3 + x}$
1. **Problem:** Find the inverse of $f(x) = \sqrt{2x + 1}$ and verify they are inverses.
2. **Step 1: Write the function as an equation:**
$$y = \sqrt{2x + 1}$$
3. **Step 2: Swap $x$ and $y$ to find the inverse:**
$$x = \sqrt{2y + 1}$$
4. **Step 3: Square both sides to eliminate the square root:**
$$x^2 = 2y + 1$$
5. **Step 4: Solve for $y$:**
$$x^2 - 1 = 2y$$
$$y = \frac{x^2 - 1}{2}$$
6. **Step 5: Write the inverse function:**
$$f^{-1}(x) = \frac{x^2 - 1}{2}$$
7. **Step 6: Verify the inverses by composition:**
- Compute $f(f^{-1}(x))$:
$$f\left(f^{-1}(x)\right) = \sqrt{2 \cdot \frac{x^2 - 1}{2} + 1} = \sqrt{x^2 - 1 + 1} = \sqrt{x^2} = |x|$$
- Compute $f^{-1}(f(x))$:
$$f^{-1}(f(x)) = \frac{(\sqrt{2x + 1})^2 - 1}{2} = \frac{2x + 1 - 1}{2} = \frac{2x}{2} = x$$
Since $f(f^{-1}(x)) = |x|$ and $f^{-1}(f(x)) = x$, the functions are inverses on the domain where $x \geq 0$ for $f^{-1}$.
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**Extra Practice Part 1:**
Given the table:
| $t$ (hours) | 0 | 6 | 12 | 18 | 24.5 | 30.5 |
|-------------|---|---|----|----|-------|-------|
| $H(t)$ (°F) | 212 | 207 | 190 | 164 | 149 | 143 |
**a. Find $H(18)$ and interpret:**
1. From the table, $H(18) = 164$.
2. Interpretation: After 18 hours, the temperature of the hot liquid in the thermos is 164 degrees Fahrenheit.
**b. Find $H^{-1}(164)$ and interpret:**
1. From the table, $H(t) = 164$ at $t = 18$ hours.
2. Interpretation: It takes 18 hours for the hot liquid to cool down to 164 degrees Fahrenheit.
**c. Find $H^{-1}(212)$ and interpret:**
1. From the table, $H(t) = 212$ at $t = 0$ hours.
2. Interpretation: The initial temperature of the hot liquid when placed in the thermos is 212 degrees Fahrenheit.
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**Problem 2: Find the inverse of $f(x) = \sqrt{3 + x}$**
1. Write the function as an equation:
$$y = \sqrt{3 + x}$$
2. Swap $x$ and $y$:
$$x = \sqrt{3 + y}$$
3. Square both sides:
$$x^2 = 3 + y$$
4. Solve for $y$:
$$y = x^2 - 3$$
5. Write the inverse function:
$$f^{-1}(x) = x^2 - 3$$