To determine the properties of the function \( h(x) = 6^x - 4 \), we need to analyze its domain, range, and asymptote.
### Domain
The function \( h(x) = 6^x - 4 \) involves an exponential term \( 6^x \). Exponential functions like \( 6^x \) are defined for all real numbers because you can raise 6 to any real number power. Therefore, the domain of \( h(x) \) is:
[tex]\[ \{x \mid x \text{ is a real number}\} \][/tex]
### Range
To find the range of \( h(x) = 6^x - 4 \), consider the properties of the exponential function \( 6^x \).
- The expression \( 6^x \) is always positive for any real number \( x \). Specifically, \( 6^x > 0 \).
- Subtracting 4 from any positive number still yields a number greater than \(-4\).
So, \( 6^x - 4 > -4 \). Therefore, the range of \( h(x) \) is:
[tex]\[ \{y \mid y > -4\} \][/tex]
### Asymptote
The function \( h(x) = 6^x - 4 \) is an exponential function shifted down by 4 units. To find the horizontal asymptote, consider the behavior of \( h(x) \) as \( x \) approaches negative infinity:
- As \( x \to -\infty \), \( 6^x \) approaches 0 since raising 6 to a very large negative number produces a value close to 0.
- Hence, \( 6^x - 4 \) approaches \( -4 \).
So, the horizontal asymptote of \( h(x) \) is:
[tex]\[ y = -4 \][/tex]
Based on this analysis, the correct answer is:
- Domain: \(\{x \mid x \text{ is a real number}\}\)
- Range: \(\{y \mid y > -4\}\)
- Asymptote: \(y = -4\)
So the correct selection is:
- Domain: [tex]\(\{x \mid x \text{ is a real number}\}\)[/tex]; Range: [tex]\(\{y \mid y > -4\}\)[/tex]; Asymptote: [tex]\(y = -4\)[/tex]
