Select the correct answer.
Which function is increasing on the interval $(-\infty, \infty)$?
A. $g(x)=-4\left(2^x\right)$
B. $j(x)=x^2+8 x+1$
C. $h(x)=2^x-1$
D. $f(x)=-3 x+7$
Answer (2)
Analyze each function to determine if it is increasing on the interval ( − ∞ , ∞ ) .
g ( x ) = − 4 ( 2 x ) is decreasing.
j ( x ) = x 2 + 8 x + 1 is a quadratic function and is not increasing on the entire interval.
h ( x ) = 2 x − 1 is increasing.
f ( x ) = − 3 x + 7 is decreasing.
The function that is increasing on the interval ( − ∞ , ∞ ) is h ( x ) = 2 x − 1 .
Explanation
Analyze each function We need to determine which of the given functions is increasing on the interval ( − ∞ , ∞ ) . Let's analyze each function:
A. g ( x ) = − 4 ( 2 x ) : This is an exponential function multiplied by a negative constant. Exponential functions 2 x are increasing, but multiplying by − 4 reflects the function across the x-axis, making it a decreasing function.
B. j ( x ) = x 2 + 8 x + 1 : This is a quadratic function. Quadratic functions are parabolas, which have a vertex. To the left of the vertex, the function decreases, and to the right, it increases (or vice versa if the parabola opens downward). Thus, a quadratic function is never increasing on the entire interval ( − ∞ , ∞ ) . To find the vertex, we can use the formula x = − b / ( 2 a ) , where a = 1 and b = 8 . So, x = − 8/ ( 2 ∗ 1 ) = − 4 . The vertex is at x = − 4 . The parabola opens upwards since 0"> a > 0 , so the function decreases for x < − 4 and increases for -4"> x > − 4 .
C. h ( x ) = 2 x − 1 : This is an exponential function. Exponential functions of the form a x where 1"> a > 1 are increasing. Subtracting 1 from the function only shifts the graph down by 1 unit, but it does not change the increasing nature of the function. Therefore, this function is increasing on ( − ∞ , ∞ ) .
D. f ( x ) = − 3 x + 7 : This is a linear function with a slope of − 3 . Since the slope is negative, the function is decreasing on ( − ∞ , ∞ ) .
Determine the increasing function Based on the analysis, the only function that is increasing on the interval ( − ∞ , ∞ ) is h ( x ) = 2 x − 1 .
Select the correct option The correct answer is C.
Examples
Imagine you're tracking the growth of a bacteria colony. If the number of bacteria doubles every hour, this growth can be modeled by an increasing exponential function, similar to the one in this problem. Understanding increasing functions helps predict how the colony will expand over time, which is crucial for controlling its spread or utilizing its properties in a lab setting. This concept is also applicable in finance, where compound interest leads to exponential growth of investments.
The only function that is increasing on the interval ( − ∞ , ∞ ) is h ( x ) = 2 x − 1 . The other functions are either decreasing or not increasing throughout the entire interval. Thus, the correct answer is C.
;
