Question
Summa'ze the pertinent information oblained by applyirg the grashing slrategy and skeldi fie giaph of \( y=f(x) \). \[ f(x)=\ln (x+8)-1 \] Select the correct choicu below and, if nacessary, fill in the anewer box fo complete your croice. A. The domain olf is \( (-8, \infty) \) (Type your answer in inlerval rolation. Usu a comma to separalo aniswers as needed.) B. The comain of \( f \) is empty. Select the correct choice balow and, if nocessary, filt in the unswer box to complete your choica A. The \( x \) intercepl(s) of \( f \) is \( x=e-\theta \). (Typo an exacl answer. Use a comma to separate answers as neoded.) B. The function 1 has no \( x \)-inlercepts Select the correct choice below and, if nocessary, fill in the answer box to corplate your chore A. Ihe \( y \)-inlercepl oll is \( y= \) \( \square \) (Type an exact answer, Uso a comma to soparalo answers as noedod.) B. The funcion thas no y-intercept
Ask by Graham Matthews. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**Summary:**
- **Domain:** All real numbers greater than \(-8\), written as \( (-8, \infty) \).
- **\(x\)-Intercept:** At \( x = e - 8 \).
- **\(y\)-Intercept:** At \( y = \ln(8) - 1 \).
Solution
**1. Domain**
The function is
\[
f(x)=\ln(x+8)-1.
\]
Since the logarithm is defined only when its argument is positive, we require
\[
x+8>0.
\]
Subtracting \(8\) from both sides gives
\[
x>-8.
\]
Thus, the domain is
\[
(-8,\infty).
\]
**2. \(x\)-Intercept**
The \(x\)-intercept(s) occur when \(f(x)=0\). Set
\[
\ln(x+8)-1=0.
\]
Add \(1\) to both sides:
\[
\ln(x+8)=1.
\]
Exponentiate both sides to remove the logarithm:
\[
x+8=e^1=e.
\]
Solve for \(x\):
\[
x=e-8.
\]
Thus, the \(x\)-intercept is at
\[
(e-8,0).
\]
**3. \(y\)-Intercept**
The \(y\)-intercept is found by evaluating \(f(x)\) at \(x=0\):
\[
f(0)=\ln(0+8)-1=\ln(8)-1.
\]
Thus, the \(y\)-intercept is
\[
(0,\,\ln(8)-1).
\]
**Final Answers:**
- **Domain:** \( (-8, \infty) \)
- **\(x\)-intercept:** \( x=e-8 \)
- **\(y\)-intercept:** \( y=\ln(8)-1 \)
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Simplify this solution Bonus Knowledge
The domain of the function \( f(x) = \ln(x + 8) - 1 \) is determined by the logarithm, which requires its argument to be positive. Thus, \( x + 8 > 0 \) leads to \( x > -8 \). Therefore, the domain is indeed \( (-8, \infty) \). The \( y \)-intercept occurs when \( x = 0 \). Plugging this into the function gives \( f(0) = \ln(0 + 8) - 1 = \ln(8) - 1 \). Hence, the \( y \)-intercept is \( y = \ln(8) - 1 \).
