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.