Question

question 7
what is the asymptote of the function \\(f(x) = -2^x - 4\\)?
\\(y = -2\\)
\\(y = -4\\)
\\(y = 4\\)

question 8
what is the asymptote of the function \\(f(x) = 2^x + 5\\)?
\\(y = -5\\)
\\(x = 5\\)
\\(y = 5\\)

Explanation:

Analyze the general form of exponential functions

An exponential function of the form \(g(x) = a \cdot b^x + c\) (where \(b > 0\) and \(b
eq 1\)) has a horizontal asymptote. As \(x\) approaches positive or negative infinity (depending on the base \(b\)), the term \(a \cdot b^x\) approaches \(0\). Therefore, the function values approach the constant value \(c\). This gives the horizontal asymptote equation:

$$y = c$$

Determine the asymptote for Question 7

For the function \(f(x) = -2^x - 4\):

• The exponential term is \(-2^x\).
• As \(x \to -\infty\), the term \(2^x \to 0\), which means \(-2^x \to 0\).
• Therefore, \(f(x) \to -4\).
• The horizontal asymptote is \(y = -4\).

Determine the asymptote for Question 8

For the function \(f(x) = 2^x + 5\):

• The exponential term is \(2^x\).
• As \(x \to -\infty\), the term \(2^x \to 0\).
• Therefore, \(f(x) \to 5\).
• The horizontal asymptote is \(y = 5\).

Answer:

Question 7

• y = -2
• y = -4 (Correct answer)
• y = 4

Question 8

• y = -5
• x = 5
• y = 5 (Correct answer)