Answer :
To determine the range of values that the functions \( h(x) = 4x + 4 \) and \( k(x) = 4^x + 3 \) have in common, we need to first analyze the ranges of each function individually.
1. Range of \( h(x) = 4x + 4 \):
- This function is a linear equation with a slope of 4 and a y-intercept of 4.
- Since it is a linear function with a non-zero slope, it can take on all real values as \( x \) varies over all real numbers.
- Therefore, the range of \( h(x) \) is \( (-\infty, \infty) \).
2. Range of \( k(x) = 4^x + 3 \):
- This function is an exponential equation shifted upward by 3 units.
- The base of the exponential part, \( 4^x \), is always positive and increases as \( x \) increases.
- The smallest value \( 4^x \) can approach, as \( x \) approaches negative infinity, is 0 but it never reaches 0.
- Therefore, the smallest value \( k(x) \) approaches is \( 0 + 3 = 3 \), but again, it never actually reaches 3.
- As \( x \) increases, \( 4^x \) grows exponentially, so the range extends to infinity.
- Hence, the range of \( k(x) \) is \( (3, \infty) \).
3. Overlapping Range:
- We now determine the intersection of the two ranges found above.
- The range of \( h(x) \) is \( (-\infty, \infty) \).
- The range of \( k(x) \) is \( (3, \infty) \).
- The intersection of \( (-\infty, \infty) \) and \( (3, \infty) \) is \( (3, \infty) \).
Thus, the range values that the two functions have in common are \((3, \infty)\). Therefore, the statement that accurately describes all the range values the two graphs have in common is:
[tex]\((3, \infty)\)[/tex].
1. Range of \( h(x) = 4x + 4 \):
- This function is a linear equation with a slope of 4 and a y-intercept of 4.
- Since it is a linear function with a non-zero slope, it can take on all real values as \( x \) varies over all real numbers.
- Therefore, the range of \( h(x) \) is \( (-\infty, \infty) \).
2. Range of \( k(x) = 4^x + 3 \):
- This function is an exponential equation shifted upward by 3 units.
- The base of the exponential part, \( 4^x \), is always positive and increases as \( x \) increases.
- The smallest value \( 4^x \) can approach, as \( x \) approaches negative infinity, is 0 but it never reaches 0.
- Therefore, the smallest value \( k(x) \) approaches is \( 0 + 3 = 3 \), but again, it never actually reaches 3.
- As \( x \) increases, \( 4^x \) grows exponentially, so the range extends to infinity.
- Hence, the range of \( k(x) \) is \( (3, \infty) \).
3. Overlapping Range:
- We now determine the intersection of the two ranges found above.
- The range of \( h(x) \) is \( (-\infty, \infty) \).
- The range of \( k(x) \) is \( (3, \infty) \).
- The intersection of \( (-\infty, \infty) \) and \( (3, \infty) \) is \( (3, \infty) \).
Thus, the range values that the two functions have in common are \((3, \infty)\). Therefore, the statement that accurately describes all the range values the two graphs have in common is:
[tex]\((3, \infty)\)[/tex].
