Find the x- and y-intercepts of the given

f(x)=3^(x+1)−4

To find the **x- and y-intercepts** of the function \( f(x) = 3^{x+1} - 4 \), follow these steps:

### 1. **Finding the y-intercept:**

The **y-intercept** occurs where \( x = 0 \).

\[
f(0) = 3^{0+1} - 4 = 3^1 - 4 = 3 - 4 = -1
\]

Thus, the **y-intercept** is at \( (0, -1) \).

### 2. **Finding the x-intercept:**

The **x-intercept** occurs where \( f(x) = 0 \). Set the function equal to 0 and solve for \( x \):

\[
0 = 3^{x+1} - 4
\]
\[
3^{x+1} = 4
\]
Now, take the natural logarithm (or logarithm base 3) of both sides:

\[
x + 1 = \log_3(4)
\]
Convert this to a natural log if needed:

\[
x + 1 = \frac{\ln(4)}{\ln(3)}
\]
\[
x = \frac{\ln(4)}{\ln(3)} - 1
\]

Calculating the values:

\[
x \approx \frac{1.386}{1.099} - 1 \approx 1.261 - 1 \approx 0.261
\]

Thus, the **x-intercept** is approximately \( (0.261, 0) \).

### Summary:
- The **y-intercept** is at \( (0, -1) \).
- The **x-intercept** is approximately at \( (0.261, 0) \).

Would you like further explanation or clarification on any of the steps?

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Here are 5 related questions to explore further:
1. How do you graph exponential functions like \( f(x) = 3^{x+1} - 4 \)?
2. What is the general form of an exponential function?
3. How do logarithmic properties help in solving equations like \( 3^{x+1} = 4 \)?
4. How do transformations affect the intercepts of exponential functions?
5. What is the domain and range of \( f(x) = 3^{x+1} - 4 \)?

**Tip:** When solving for x-intercepts in exponential equations, logarithms are often the best tool to simplify the expression.

Find the x- and y-intercepts of the given f(x) = 3^(x+1) - 4

Learn how to find the x- and y-intercepts of the exponential function f(x) = 3^(x+1) - 4. This solution covers key concepts such as intercepts, logarithmic conversions, and exponential functions. Suitable for students in Grades 10-12.

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