Consider the functions. f(x) =-x^2-2x+3 and g(x) =-2×2^x-1(both exponential to 2) +1

Question

1)Show how g can be simplified to g(x) =-2^x +1. 2) Draw a neat graph of both functions on the same set of axes. Indicate the intercept with axis and any asymptotes. 3) Determine the equation for the axis of symmetry of f. 4) Determine the range of f. 5) Determine the range of g. 6) Use the graph th determine the value of f(0)--(0). 7) What are the values of k, for which - x^2-2x+k=0 will have no real roots?

Answer 1

(1) g(x)=-2×2ˣ⁻¹+1, -(2¹⁺ˣ⁻¹)+1=-2ˣ+1.

(2) f(x)=-x²-2x+3 is blue and g(x) is red.

The x-intercepts are the roots of f(x) and the solution to -x²-2x+3=0=-(x+3)(x-1).

(3) Axis of symmetry for f is x=-1 (green line). Asymptote passes vertically through the vertex of the parabola.

(4) Range of f is (-∞,4].

(5) Range of g is (-∞,1).

(6) I guess you mean f(0)-g(0)=3-0=3.

(7) Imagine pulling down the blue curve so that it’s just below the x-axis (so there can be no x-intercepts and hence no real solution). The “hump” or vertex must be below the x-axis which means the y intercept at 3 must be lower than y=-1. The constant 3 is the current y-intercept and it needs to move more than 4 units down so that the vertex is below the x-axis, so k<3-4, k<-1.