Question

identify the key features of the exponential function f(x) = 6^x and its graph by completing each sentence. the domain is \boxed{}. the range is y > \boxed{}. the asymptote is y = \boxed{}. the y - intercept is \boxed{}.

Explanation:

Step1: Find the domain of \( f(x) = 6^x \)

For an exponential function \( a^x \) (where \( a>0, a
eq1 \)), \( x \) can be any real number. So the domain is all real numbers, which is \( (-\infty, \infty) \) or \( \mathbb{R} \).

Step2: Find the range of \( f(x) = 6^x \)

Since \( 6>1 \), as \( x \to -\infty \), \( 6^x \to 0 \), and as \( x \to \infty \), \( 6^x \to \infty \). Also, \( 6^x>0 \) for all real \( x \). So the range is \( y > 0 \).

Step3: Find the asymptote of \( f(x) = 6^x \)

As \( x \to -\infty \), \( 6^x \to 0 \), so the horizontal asymptote is \( y = 0 \).

Step4: Find the y - intercept of \( f(x) = 6^x \)

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0)=6^0 = 1 \). So the y - intercept is \( 1 \).

Answer:

• The domain is \( (-\infty, \infty) \) (or all real numbers).
• The range is \( y>0 \).
• The asymptote is \( y = 0 \).
• The y - intercept is \( 1 \).