QUESTION 3 Given \( f(x)=-(x+2)^{2}+6 \) and \( g(x)=2^{-x}+1 \). 3.1 Draw graphs of \( f \) and \( g \) on the same set of axes. Clearty show the intercepts with both axes, as well as the asymptote(s) where appliable. 3.2 Wriue down the value(s) of \( f \) if \( f(x)=1 \) tans: 3.2 .1 equal roots. 3.22 one root equal to 0 . 3.3 Write down the equation of the asymptote of \( h \) if \( h(x)=g(x)+1 \)
Upstudy AI Solution
Answer
Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain 
Simplify this solution Mind Expander
The function \( f(x) = -(x+2)^2 + 6 \) describes a downward-opening parabola with its vertex at the point (-2, 6). The intercepts can be found: the y-intercept occurs when \( x=0 \), giving \( f(0) = 2 \); the x-intercepts occur by setting \( f(x) = 0 \), leading to roots at \( x = -4 \) and \( x = 0 \). In contrast, \( g(x) = 2^{-x}+1 \) is an exponential function with a horizontal asymptote at \( y = 1 \) and intercepts at \( (0, 2) \). In terms of applications, the intersection points of \( f(x) \) and \( g(x) \) can represent various real-world scenarios, such as the balance between two competing trends, such as revenue versus cost, in a business model. Finding \( x \) values where these functions match could signify break-even points for optimizing profits or costs. A useful tip is to always sketch or graph functions before analyzing their interactions to visualize any intersections or asymptotic behaviors, which often helps avoid mistakes in determining roots or intercepts.
