Use the function h(x)=3(.83^3x) for the following problems:
1.) ___ This is (a) an exponential growth function (b) a logarithmic function (c) an exponential decay function (d) a function using natural logarithms
2.) ___ The y-intercept (a) is at y≈.57 (b) is at y=.83 (c) is at x=3 (d) does not exist (e) none of the above
3.) ___ The x-intercept (a) is at x=0 (b) is at x≈.57 (c) is at y=3 (d) does not exist (e) none of the above
4.) The end behavior of this function is best described as:
(a) h(x)-> ∞ as x->- ∞; h(x)->0 as x-> ∞
(b) h(x)->0 as x->- ∞; h(x)-> ∞ as x-> ∞
(c) h(x)-> - ∞ as x->- ∞; h(x)->0 as x-> ∞
(d) h(x)->0 as x->- ∞; h(x)->- ∞ as x-> ∞
For #5 use the following equation: k(x)=(x-2)^2+2
5.) Choose all the true statements:
(a) k(x)>h(x) for x≤0
(b) as x-> ∞, k(x) > h(x)
(c) as x->-∞, h(x) > k(x)
(d) k(x)>h(x) for |x|≤1
(e) h(x)>k(x) for -1 <x≤5
(f) as x-> 2, h(x)>k(x)
1) This is an exponential function
2) Y-intercept is determined by y-value when x = 0. So, `y = h(0) = 3*0.83^(3*0)=3`
The y-intercept of h(x) is y = 3, but there is no answer choice like this. So it is either x = 3 which was meant to be typed "y = 3", or "none of the above".
3) X-intercept does not exist, because the graph of h(x) never intersects x-axis. h(x) is not 0 for any x.
4) The end behavior: when x is approaching positive infinity, that is, becoming very large, h(x) is also becoming very large and approaches infinity. When x is approaching negative infinity, h(x) is becoming very small and approaches 0, but never reaches it. So the answer is b).
4) Correction: the answer is a). Since the base of exponent is less than 1, h(x) approaches 0 when x is very large (approaches positive infinity) and h(x) approaches positive infinity when x is large and negative (approaches negative infinity.)