Rowe Horton
12/24/2023 · Primary School
If \( g(x)=7+x+e^{x} \), find \( g^{-1}(8) \) \( g^{-1}(8)= \)
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Tutor-Verified Answer
Step-by-step Solution
To find \( g^{-1}(8) \), we need to solve the equation \( g(x) = 8 \) for \( x \). Given the function \( g(x) = 7 + x + e^x \), we set up the equation:
\[
7 + x + e^x = 8
\]
Subtracting 7 from both sides gives:
\[
x + e^x = 1
\]
Now, we need to solve the equation \( x + e^x = 1 \). We can rearrange it to:
\[
e^x = 1 - x
\]
Next, we will analyze the function \( h(x) = e^x + x \) and find where it equals 1. We can evaluate \( h(x) \) at a few points:
1. For \( x = 0 \):
\[
h(0) = e^0 + 0 = 1 + 0 = 1
\]
2. For \( x = -1 \):
\[
h(-1) = e^{-1} - 1 \approx 0.3679 - 1 \approx -0.6321
\]
3. For \( x = 1 \):
\[
h(1) = e^1 + 1 \approx 2.7183 + 1 \approx 3.7183
\]
From these evaluations, we see that \( h(0) = 1 \), and since \( h(x) \) is a continuous and strictly increasing function (as both \( e^x \) and \( x \) are increasing), we conclude that \( h(x) = 1 \) has a unique solution at \( x = 0 \).
Thus, we have:
\[
g(0) = 7 + 0 + e^0 = 7 + 0 + 1 = 8
\]
This means:
\[
g^{-1}(8) = 0
\]
Therefore, the final answer is:
\[
\boxed{0}
\]
Quick Answer
\( g^{-1}(8) = 0 \)
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