Answer :
To determine the domain, range, and asymptote of the function [tex]\( h(x) = 6^x - 4 \)[/tex], we will analyze each aspect step-by-step.
### 1. Domain
The domain of a function refers to the set of all possible input values (x-values) that the function can accept. For the exponential function [tex]\( 6^x - 4 \)[/tex]:
- Exponential functions are defined for all real numbers.
- Thus, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
In set notation, this is expressed as:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
### 2. Range
The range of a function refers to the set of all possible output values (y-values) that the function can produce.
- Consider the exponential part [tex]\( 6^x \)[/tex]. Since [tex]\( 6^x \)[/tex] is an exponential function with base greater than 1, it produces positive values for all real numbers [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 6^x \)[/tex] approaches 0.
- Therefore, [tex]\( 6^x - 4 \)[/tex] will approach [tex]\( -4 \)[/tex] but never actually reach [tex]\( -4 \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( 6^x \rightarrow \infty \)[/tex], so [tex]\( 6^x - 4 \rightarrow \infty \)[/tex].
Thus, the range of [tex]\( h(x) = 6^x - 4 \)[/tex] is:
[tex]\[ \{ y \mid y > -4 \} \][/tex]
### 3. Asymptote
An asymptote is a line that the graph of a function approaches but never actually touches.
- For the function [tex]\( h(x) = 6^x - 4 \)[/tex], as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 6^x \)[/tex] approaches 0, making [tex]\( h(x) \)[/tex] approach [tex]\( -4 \)[/tex].
- Therefore, the horizontal asymptote is:
[tex]\[ y = -4 \][/tex]
Based on this analysis, the correct descriptions are:
- Domain: [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex]
- Range: [tex]\(\{ y \mid y > -4 \}\)[/tex]
- Asymptote: [tex]\( y = -4 \)[/tex]
Thus, the correct option is:
[tex]\[ \text{domain: } \{ x \mid x \text{ is a real number} \}; \text{ range: } \{ y \mid y > -4 \}; \text{ asymptote: } y = -4 \][/tex]
### 1. Domain
The domain of a function refers to the set of all possible input values (x-values) that the function can accept. For the exponential function [tex]\( 6^x - 4 \)[/tex]:
- Exponential functions are defined for all real numbers.
- Thus, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
In set notation, this is expressed as:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
### 2. Range
The range of a function refers to the set of all possible output values (y-values) that the function can produce.
- Consider the exponential part [tex]\( 6^x \)[/tex]. Since [tex]\( 6^x \)[/tex] is an exponential function with base greater than 1, it produces positive values for all real numbers [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 6^x \)[/tex] approaches 0.
- Therefore, [tex]\( 6^x - 4 \)[/tex] will approach [tex]\( -4 \)[/tex] but never actually reach [tex]\( -4 \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( 6^x \rightarrow \infty \)[/tex], so [tex]\( 6^x - 4 \rightarrow \infty \)[/tex].
Thus, the range of [tex]\( h(x) = 6^x - 4 \)[/tex] is:
[tex]\[ \{ y \mid y > -4 \} \][/tex]
### 3. Asymptote
An asymptote is a line that the graph of a function approaches but never actually touches.
- For the function [tex]\( h(x) = 6^x - 4 \)[/tex], as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 6^x \)[/tex] approaches 0, making [tex]\( h(x) \)[/tex] approach [tex]\( -4 \)[/tex].
- Therefore, the horizontal asymptote is:
[tex]\[ y = -4 \][/tex]
Based on this analysis, the correct descriptions are:
- Domain: [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex]
- Range: [tex]\(\{ y \mid y > -4 \}\)[/tex]
- Asymptote: [tex]\( y = -4 \)[/tex]
Thus, the correct option is:
[tex]\[ \text{domain: } \{ x \mid x \text{ is a real number} \}; \text{ range: } \{ y \mid y > -4 \}; \text{ asymptote: } y = -4 \][/tex]
