which of the statements about the graph of the function y = 2^x are true? check all of the boxes that apply…

which of the statements about the graph of the function y = 2^x are true? check all of the boxes that apply. the domain is all real numbers x because the exponent of 2 can be any real number when the x - values increases by 1 unit, the y - value multiplies by 2. the y - intercept is (0, 1). the graph never goes below the x - axis because powers of 2 are never negative. the range is all real numbers. done

which of the statements about the graph of the function y = 2^x are true? check all of the boxes that apply. the domain is all real numbers x because the exponent of 2 can be any real number when the x - values increases by 1 unit, the y - value multiplies by 2. the y - intercept is (0, 1). the graph never goes below the x - axis because powers of 2 are never negative. the range is all real numbers. done

Answer

Explanation:

Step1: Analyze domain

For the exponential - function (y = 2^{x}), the exponent (x) can be any real number. So the domain is all real numbers. The statement "The domain is all real numbers (x) because the exponent of 2 can be any real number" is true.

Step2: Analyze the change of (y) with (x)

If (x) changes from (x_1) to (x_1 + 1), then (y_1=2^{x_1}) and (y_2 = 2^{x_1 + 1}=2\times2^{x_1}). So when the (x) - values increases by 1 unit, the (y) - value multiplies by 2. This statement is true.

Step3: Analyze (y) - intercept

To find the (y) - intercept, set (x = 0). Then (y=2^{0}=1). So the (y) - intercept is ((0,1)). This statement is true.

Step4: Analyze the position of the graph

Since (2^{x}>0) for all real - valued (x) (because any non - zero number to a real power is non - negative, and (2>0)), the graph of (y = 2^{x}) never goes below the (x) - axis. This statement is true.

Step5: Analyze range

The function (y = 2^{x}) has a range of (y>0) (not all real numbers). As (x\to-\infty), (y\to0) but (y\neq0), and as (x\to+\infty), (y\to+\infty). So the statement "The range is all real numbers" is false.

Answer:

The domain is all real numbers (x) because the exponent of 2 can be any real number. When the (x) - values increases by 1 unit, the (y) - value multiplies by 2. The (y) - intercept is ((0,1)). The graph never goes below the (x) - axis because powers of 2 are never negative.