To determine whether the function [tex]\( f(x) = 7^x + 3 \)[/tex] represents exponential growth, decay, or neither, we need to analyze the form and behavior of the function.
1. Understanding the Base of the Exponential Function:
- In the function [tex]\( f(x) = 7^x + 3 \)[/tex], the term [tex]\( 7^x \)[/tex] is an exponential expression with a base of 7.
- The base of an exponential function dictates whether it exhibits growth or decay. When the base [tex]\( b \)[/tex] of [tex]\( b^x \)[/tex] is greater than 1, the function [tex]\( b^x \)[/tex] shows exponential growth. Conversely, if [tex]\( 0 < b < 1 \)[/tex], the function shows exponential decay.
2. Evaluating the Given Function:
- Here, the base of the exponent is 7, which is greater than 1.
- Therefore, the term [tex]\( 7^x \)[/tex] by itself represents exponential growth.
3. Effect of the Constant Term:
- The function [tex]\( f(x) = 7^x + 3 \)[/tex] includes an additional constant term +3.
- Adding a constant to an exponential function shifts the graph vertically but does not affect the growth or decay nature of the exponential term.
4. Conclusion:
- Since [tex]\( 7^x \)[/tex] grows exponentially as [tex]\( x \)[/tex] increases and the constant term [tex]\( +3 \)[/tex] merely shifts the graph upward, the overall function [tex]\( f(x) = 7^x + 3 \)[/tex] represents exponential growth.
Thus, the function [tex]\( f(x) = 7^x + 3 \)[/tex] indeed represents exponential growth, leading us to conclude that the correct answer is:
D) Exponential growth
