Let's analyze the function [tex]\( y = 2^x \)[/tex] step-by-step and check each of the given statements:
1. The domain is all real numbers [tex]\( x \)[/tex] because the exponent of 2 can be any real number:
- The domain of a function is the set of all possible input values (x-values). For the function [tex]\( y = 2^x \)[/tex], [tex]\( x \)[/tex] can indeed be any real number, since you can raise 2 to any real number power.
- This statement is true.
2. When the [tex]\( x \)[/tex]-values increase by 1 unit, the [tex]\( y \)[/tex] value multiplies by 2:
- Consider the function [tex]\( y = 2^x \)[/tex]. If [tex]\( x \)[/tex] increases by 1 unit, from [tex]\( x \)[/tex] to [tex]\( x+1 \)[/tex], then
[tex]\[
y = 2^{x+1} = 2^x \cdot 2
\][/tex]
Therefore, the [tex]\( y \)[/tex]-value indeed multiplies by 2.
- This statement is true.
3. The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]:
- The [tex]\( y \)[/tex]-intercept of a function is the point where the graph crosses the [tex]\( y \)[/tex]-axis, i.e., the point where [tex]\( x = 0 \)[/tex].
[tex]\[
y = 2^0 = 1
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 1) \)[/tex].
- This statement is true.
4. The graph never goes below the [tex]\( x \)[/tex]-axis because powers of 2 are never negative:
- For the function [tex]\( y = 2^x \)[/tex], no matter what real number [tex]\( x \)[/tex] is, [tex]\( 2^x \)[/tex] is always positive (greater than zero). It can never be negative or zero.
- This statement is true.
5. The range is all real numbers:
- The range of a function is the set of all possible output values (y-values). For [tex]\( y = 2^x \)[/tex], the outputs are all positive real numbers, but never zero or negative. The range is therefore [tex]\( (0, \infty) \)[/tex].
- This statement is false.
Thus, the checked boxes should be:
- The domain is all real numbers [tex]\( x \)[/tex] because the exponent of 2 can be any real number.
- When the [tex]\( x \)[/tex]-values increase by 1 unit, the [tex]\( y \)[/tex] value multiplies by 2.
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0, 1) \)[/tex].
- The graph never goes below the [tex]\( x \)[/tex]-axis because powers of 2 are never negative.
