To find the other solution of the equation
\[ \sqrt{3x+3} = e^x, \]
we can rewrite it to identify the points where the two sides of the equation intersect.
Let's define the functions:
- \( f(x) = \sqrt{3x + 3} \)
- \( g(x) = e^x \)
We want to find the intersections of these two functions by solving \( f(x) = g(x) \).
Analyzing the Functions
-
Domain of \( f(x) \):
- The expression under the square root, \( 3x + 3 \), must be non-negative: \[ 3x + 3 \geq 0 \implies x \geq -1. \] Therefore, the domain of \( f(x) \) is \( x \geq -1 \).
-
Behavior of \( f(x) \) and \( g(x) \):
- For \( f(x) = \sqrt{3x + 3} \):
- At \( x = -1 \), \( f(-1) = \sqrt{3(-1) + 3} = \sqrt{0} = 0 \).
- As \( x \to \infty \), \( f(x) \) increases without bound (as it behaves like \( \sqrt{3x} \)).
- For \( g(x) = e^x \):
- At \( x = -1 \), \( g(-1) = e^{-1} \approx 0.3679 \).
- As \( x \to \infty \), \( g(x) \) also increases without bound.
- For \( f(x) = \sqrt{3x + 3} \):
Using a Graphing Calculator
To find the intersection points, we will graph both functions \( f(x) = \sqrt{3x + 3} \) and \( g(x) = e^x \).
Upon graphing, we see that:
- One intersection occurs near \( x \approx -1 \), which we are given.
- The other intersection appears to be at a larger value of \( x \).
Estimate the Intersection
By analyzing the graph using a calculator or graphing software, we can find the values of \( x \) where \( f(x) \) and \( g(x) \) intersect.
After observing the graph:
- Move along the \( x \)-axis to find the point where both functions converge again (apart from \( x \approx -1 \)).
- You should find the second intersection occurs around \( x \approx 1.3 \).
Thus, rounding this to the nearest tenth, the other solution is:
\[ \boxed{1.3} \]