Question

what are the domain, range, and asymptote of $h(x) = 6^x - 4$?
domain: ${x \mid x \text{ is a real number}\\}$; range: ${y \mid y > -4\\}$; asymptote: $y = -4$
domain: ${x \mid x > -4\\}$; range: ${y \mid y \text{ is a real number}\\}$; asymptote: $y = 4$
domain: ${x \mid x \text{ is a real number}\\}$; range: ${y \mid y > 4\\}$; asymptote: $y = 4$
domain: ${x \mid x > 4\\}$; range: ${y \mid y \text{ is a real number}\\}$; asymptote: $y = -4$

Explanation:

Step1: Analyze the function \( h(x) = 6^x - 4 \)

The function \( 6^x \) is an exponential function. For any real number \( x \), \( 6^x \) is defined. So the domain of \( 6^x \) is all real numbers. When we subtract 4, the domain remains all real numbers because subtracting a constant doesn't restrict the input values. So the domain of \( h(x) \) is \( \{x \mid x \text{ is a real number}\} \).

Step2: Determine the range of \( h(x) \)

The range of \( 6^x \) is \( \{y \mid y > 0\} \) because an exponential function with a positive base (greater than 1) always gives positive output values. Now, for \( h(x) = 6^x - 4 \), we subtract 4 from each output of \( 6^x \). So the range of \( h(x) \) will be \( \{y \mid y > 0 - 4\} = \{y \mid y > -4\} \).

Step3: Find the asymptote of \( h(x) \)

The horizontal asymptote of \( 6^x \) is \( y = 0 \) (as \( x \to -\infty \), \( 6^x \to 0 \)). When we have \( h(x) = 6^x - 4 \), the graph of \( 6^x \) is shifted down by 4 units. So the horizontal asymptote also shifts down by 4 units, resulting in \( y = -4 \).

Now we check the options:

• First option: domain is all real numbers, range \( y > -4 \), asymptote \( y = -4 \) – matches our analysis.
• Second option: domain \( x > -4 \) (incorrect, domain of exponential is all real numbers), range all real numbers (incorrect, range is \( y > -4 \)), asymptote \( y = 4 \) (incorrect) – eliminate.
• Third option: range \( y > 4 \) (incorrect, should be \( y > -4 \)), asymptote \( y = 4 \) (incorrect) – eliminate.
• Fourth option: domain \( x > 4 \) (incorrect), range all real numbers (incorrect) – eliminate.

Answer:

domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > -4\}\); asymptote: \(y = -4\)