To determine which function is increasing on the interval [tex]\((-\infty, \infty)\)[/tex], let's analyze each option step-by-step:
### Function A: [tex]\( j(x) = x^2 + 8x + 1 \)[/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 1 \)[/tex].
1. The leading coefficient [tex]\( a = 1 \)[/tex] is positive.
2. Quadratic functions with positive leading coefficients open upwards, forming a parabola.
3. Such parabolas initially decrease until they reach the vertex (the minimum point) and then increase.
Since [tex]\( j(x) \)[/tex] first decreases and then increases, it is not increasing on the entire interval [tex]\((-\infty, \infty)\)[/tex].
### Function B: [tex]\( f(x) = -3x + 7 \)[/tex]
This is a linear function of the form [tex]\( mx + b \)[/tex], where [tex]\( m = -3 \)[/tex] and [tex]\( b = 7 \)[/tex].
1. The slope [tex]\( m = -3 \)[/tex] is negative.
2. Linear functions with negative slopes decrease as [tex]\( x \)[/tex] increases.
Since [tex]\( f(x) \)[/tex] decreases on the interval [tex]\((-\infty, \infty)\)[/tex], it is not increasing.
### Function C: [tex]\( g(x) = -4(2^x) \)[/tex]
This is an exponential function of the form [tex]\( a \cdot (b^x) \)[/tex] where [tex]\( a = -4 \)[/tex] and [tex]\( b = 2 \)[/tex].
1. The base of the exponential function is 2, which is greater than 1, meaning [tex]\( 2^x \)[/tex] itself is increasing.
2. However, multiplying by a negative constant [tex]\( -4 \)[/tex] will reverse the direction of the function, making [tex]\( g(x) \)[/tex] decrease as [tex]\( x \)[/tex] increases.
Hence, [tex]\( g(x) \)[/tex] is not increasing on the interval [tex]\((-\infty, \infty)\)[/tex].
### Function D: [tex]\( h(x) = 2^x - 1 \)[/tex]
This is another exponential function of the form [tex]\( b^x \)[/tex], where [tex]\( b = 2 \)[/tex], shifted down by 1 unit.
1. The base of the exponential function is 2, which is greater than 1.
2. Exponential functions with bases greater than 1 are always increasing as [tex]\( x \)[/tex] increases.
3. Subtracting 1 does not change the increasing nature of the exponential function; it merely shifts the graph down by 1 unit.
Therefore, [tex]\( h(x) \)[/tex] is increasing on the interval [tex]\((-\infty, \infty)\)[/tex].
### Conclusion:
The function that is increasing on the interval [tex]\((-\infty, \infty)\)[/tex] is:
[tex]\[ \boxed{h(x) = 2^x - 1} \][/tex]
So the correct answer is:
[tex]\[ \boxed{D} \][/tex]
