Question

which is the graph of \\(f(x) = 4^x\\)?

the domain of \\(f(x) = 4^x\\) is dropdown
the range of \\(f(x) = 4^x\\) is dropdown

Explanation:

Identify the correct graph

Using the Exponential Function Graphs knowledge point

• The function is \(f(x) = 4^x\), which is an exponential growth function since the base \(4 > 1\).
• As \(x \to \infty\), \(f(x) \to \infty\). As \(x \to -\infty\), \(f(x) \to 0\) (horizontal asymptote at \(y = 0\)).
• The \(y\)-intercept is \(f(0) = 4^0 = 1\).
• The first graph shows an exponential growth curve passing through \((0,1)\) and \((1,4)\). The second graph shows a parabola.
• Therefore, the first graph is the correct graph of \(f(x) = 4^x\).

Determine the domain

Using the Domain of Exponential Functions knowledge point

• The exponential function \(f(x) = 4^x\) is defined for all real numbers.
• There are no restrictions such as division by zero or square roots of negative numbers.
• Therefore, the domain is all real numbers, represented in interval notation as \((-\infty, \infty)\).

Determine the range

Using the Range of Exponential Functions knowledge point

• Since the base \(4\) is positive, \(4^x\) is always strictly greater than \(0\) for any real value of \(x\).
• As \(x \to -\infty\), \(4^x\) approaches \(0\) but never reaches it.
• Therefore, the range is all positive real numbers, represented in interval notation as \((0, \infty)\) or \(y > 0\).

Answer:

Question 1

Which is the graph of \(f(x) = 4^x\)?

• The first graph (exponential curve passing through \((0,1)\)) (Correct answer)
• The second graph (parabola centered at the origin)

Question 2

The domain of \(f(x) = 4^x\) is <blank>all real numbers</blank> (or \((-\infty, \infty)\)).

Question 3

The range of \(f(x) = 4^x\) is <blank>\(y > 0\)</blank> (or \((0, \infty)\)).