Question

identify the key features of the function from the table.

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$$\begin{array}{|c|c|} \\hline x & f(x) \\\\ \\hline -2 & \\frac{1}{4} \\\\ \\hline -1 & \\frac{1}{2} \\\\ \\hline 0 & 1 \\\\ \\hline 1 & 2 \\\\ \\hline 2 & 4 \\\\ \\hline \\end{array}$$

• the function is decreasing
• a = 1
• as x approaches negative infinity, f(x) approaches positive infinity.
• a = 2
• b = 2
• the function is increasing
• as x approaches positive infinity, f(x) approaches positive infinity.
• b = 1

Explanation:

Identify the function type

The table shows values of \(f(x)\) for integer values of \(x\):

• \(f(-2) = \frac{1}{4}\)
• \(f(-1) = \frac{1}{2}\)
• \(f(0) = 1\)
• \(f(1) = 2\)
• \(f(2) = 4\)

As \(x\) increases by \(1\), \(f(x)\) is multiplied by \(2\). This represents an exponential function of the form \(f(x) = a \cdot b^x\).

Determine parameters \(a\) and \(b\)

Using the standard exponential form \(f(x) = a \cdot b^x\):

• The \(y\)-intercept occurs at \(x = 0\), so \(f(0) = a \cdot b^0 = a = 1\).
• Using \(f(1) = 2\), we have \(1 \cdot b^1 = 2\), which gives \(b = 2\).

Thus, the parameters are \(a = 1\) and \(b = 2\).

Analyze function behavior

Let's evaluate the statements based on \(f(x) = 2^x\):

• Increasing/Decreasing: Since the base \(b = 2 > 1\), the function is strictly increasing.
• End Behavior:
• As \(x \to \infty\), \(f(x) = 2^x \to \infty\) (approaches positive infinity).
• As \(x \to -\infty\), \(f(x) = 2^x \to 0\) (approaches zero).

Match with given options

Based on our analysis, the correct features are:

• \(a = 1\)
• \(b = 2\)
• The function is increasing
• As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity.

Answer:

• The function is decreasing
• \(a = 1\) (Correct answer)
• As x approaches negative infinity, f(x) approaches positive infinity.
• a = 2
• \(b = 2\) (Correct answer)
• The function is increasing (Correct answer)
• As x approaches positive infinity, f(x) approaches positive infinity. (Correct answer)
• b = 1