To determine which function is increasing over the interval [tex]\((-∞, ∞)\)[/tex], we need to analyze the derivatives of each function.
### Function [tex]\( j(x) = x^2 + 8x + 1 \)[/tex]
1. To find if [tex]\( j(x) \)[/tex] is increasing, we need to calculate its derivative.
2. The derivative is [tex]\( j'(x) = 2x + 8 \)[/tex].
3. This derivative is positive when [tex]\( x > -4 \)[/tex] and negative when [tex]\( x < -4 \)[/tex].
4. Therefore, [tex]\( j(x) \)[/tex] is not increasing for all [tex]\( x \in (-∞, ∞) \)[/tex].
### Function [tex]\( g(x) = -4(2^x) \)[/tex]
1. To determine if [tex]\( g(x) \)[/tex] is increasing, we calculate its derivative.
2. The derivative is [tex]\( g'(x) = -4 (\ln(2)) 2^x \)[/tex].
3. Since [tex]\(-4 (\ln(2)) 2^x\)[/tex] is always negative for all [tex]\(x\)[/tex], [tex]\( g(x) \)[/tex] is always decreasing.
4. Thus, [tex]\( g(x) \)[/tex] is not increasing for all [tex]\( x \in (-∞, ∞) \)[/tex].
### Function [tex]\( f(x) = -3x + 7 \)[/tex]
1. To determine if [tex]\( f(x) \)[/tex] is increasing, we calculate its derivative.
2. The derivative is [tex]\( f'(x) = -3 \)[/tex].
3. Since [tex]\(-3\)[/tex] is a constant negative number, [tex]\( f(x) \)[/tex] is always decreasing.
4. Thus, [tex]\( f(x) \)[/tex] is not increasing for all [tex]\( x \in (-∞, ∞) \)[/tex].
### Function [tex]\( h(x) = 2^x - 1 \)[/tex]
1. To determine if [tex]\( h(x) \)[/tex] is increasing, we calculate its derivative.
2. The derivative is [tex]\( h'(x) = (\ln(2)) 2^x \)[/tex].
3. Since [tex]\(\ln(2)\)[/tex] is a positive constant and [tex]\( 2^x \)[/tex] is always positive for all [tex]\( x \)[/tex], [tex]\( (\ln(2)) 2^x \)[/tex] is always positive.
4. Thus, [tex]\( h(x) \)[/tex] is increasing for all [tex]\( x \in (-∞, ∞) \)[/tex].
Therefore, the function that is increasing on the interval [tex]\((-∞, ∞)\)[/tex] is:
[tex]\[ \boxed{h(x) = 2^x - 1} \][/tex]
The correct answer is:
[tex]\[ \text{D. } h(x) = 2^x - 1 \][/tex]
