Question

Please help me with this question

Answer 1

• This is a growth function. As x increases, y increases
• y-intercept is (0, 2.25)
• Equation of the horizontal asymptote is: y = 2
• Domain is (-∞, ∞)
• Range is y > 2
• End behavior
  As x
  `\rightarrow` -∞, f(x)
  `\rightarrow` 2
  As x
  `\rightarrow`+∞, f(x)
  `\rightarrow` ∞

Explanation:

First plot the graph. That will answer most of the questions. Plot is attached

• You can see that as x increases, y increases. Hence, a growth function
• To determine y-intercept, set x = 0 and solve for y:
  ==> 2⁰⁻² + 2 ==> 2⁻² + 2 ==> 1/4 + 2 = 2.25
• The horizontal asymptote is the horizontal line where the function value would have become as x approaches ± ∞. This is the horizontal red line in the plot and you can see that happens at y = 2
• Domain is the set of all real numbers so (-∞, ∞)
• Range is defined for all values of y > 2 as you can see from the graph
• As x approaches -∞ you can see that f(x) approaches the horizontal asymptote value of 2
• As x increases, there is no upper bound on the value of f(x) hence y tends to ∞
  

Hope that helps