Answer :
To find the domain and range of the function \( f(x) = -\log(5 - x) + 9 \), let's analyze it step by step.
### Domain
1. Argument of the Logarithm: The logarithmic function \(\log(z)\) is only defined for positive values of \( z \). Therefore, the argument \( 5 - x \) must be positive:
[tex]\[ 5 - x > 0 \][/tex]
2. Solving the Inequality: Solve this inequality for \( x \):
[tex]\[ 5 > x \quad \text{or} \quad x < 5 \][/tex]
Hence, the domain of \( f(x) \) is:
[tex]\[ x < 5 \][/tex]
### Range
1. Behavior of the Logarithmic Function:
- As \( x \) approaches \( 5 \) from the left, \( 5 - x \) approaches \( 0 \) from the positive side, which makes \(\log(5 - x)\) approach \( -\infty \).
- Therefore, \(-\log(5 - x)\) will approach \( \infty \).
2. Shift and Reflection:
- Our function \( f(x) \) is \(-\log(5 - x) + 9\). The term \( +9 \) shifts the entire function up by 9 units.
- As \(-\log(5 - x)\) approaches \( \infty \), \( f(x) \) approaches \( \infty + 9 = \infty \).
- When \( x \) is very small and negative, \( 5 - x \) is large, which makes \(\log(5 - x)\) large, leading \(-\log(5 - x)\) to be a large negative number. Thus, \( f(x) \) will approach \( 9 - \infty = -\infty \).
3. Considering All Values:
- The smallest value for \( -\log(5 - x) \) occurs as \( x \) approaches \( 5 \), and in this case, \(-\log(5 - x)\) is very large. The function will be slightly less than \( 9 \).
- There are no upper bounds for \( y \) as shown earlier, approaching \( \infty \).
Hence, the range of \( f(x) \) is:
[tex]\[ y \geq 9 \][/tex]
### Summary
- Domain: \( x < 5 \)
- Range: \( y \geq 9 \)
So, the correct option summarizing the domain and range is:
- Domain: \( x < 5 \)
- Range: [tex]\( y \geq 9 \)[/tex]
### Domain
1. Argument of the Logarithm: The logarithmic function \(\log(z)\) is only defined for positive values of \( z \). Therefore, the argument \( 5 - x \) must be positive:
[tex]\[ 5 - x > 0 \][/tex]
2. Solving the Inequality: Solve this inequality for \( x \):
[tex]\[ 5 > x \quad \text{or} \quad x < 5 \][/tex]
Hence, the domain of \( f(x) \) is:
[tex]\[ x < 5 \][/tex]
### Range
1. Behavior of the Logarithmic Function:
- As \( x \) approaches \( 5 \) from the left, \( 5 - x \) approaches \( 0 \) from the positive side, which makes \(\log(5 - x)\) approach \( -\infty \).
- Therefore, \(-\log(5 - x)\) will approach \( \infty \).
2. Shift and Reflection:
- Our function \( f(x) \) is \(-\log(5 - x) + 9\). The term \( +9 \) shifts the entire function up by 9 units.
- As \(-\log(5 - x)\) approaches \( \infty \), \( f(x) \) approaches \( \infty + 9 = \infty \).
- When \( x \) is very small and negative, \( 5 - x \) is large, which makes \(\log(5 - x)\) large, leading \(-\log(5 - x)\) to be a large negative number. Thus, \( f(x) \) will approach \( 9 - \infty = -\infty \).
3. Considering All Values:
- The smallest value for \( -\log(5 - x) \) occurs as \( x \) approaches \( 5 \), and in this case, \(-\log(5 - x)\) is very large. The function will be slightly less than \( 9 \).
- There are no upper bounds for \( y \) as shown earlier, approaching \( \infty \).
Hence, the range of \( f(x) \) is:
[tex]\[ y \geq 9 \][/tex]
### Summary
- Domain: \( x < 5 \)
- Range: \( y \geq 9 \)
So, the correct option summarizing the domain and range is:
- Domain: \( x < 5 \)
- Range: [tex]\( y \geq 9 \)[/tex]
