Question

r^(2). Do as indicated. a. Find all values of m such that g(x)={(x+1 if x<=m,),(x²) if x>m,):} is continuous everywhere.

Answer 1

To find the values of m for the function g(x) to be continuous, we must solve the quadratic equation m^2 - m - 1 = 0 and find the values of m where the limits from the left and right are equal as well as equal to g(m).

Step-by-step explanation:

To determine the values of m such that the function g(x) is continuous everywhere, we need to ensure g(x) is continuous at m. Since one piece of g(x) is defined for x ≤ m and the other for x > m, the point of potential discontinuity is at x = m.

For g(x) to be continuous at x = m:

• The function must not have a jump discontinuity, or in other words, the limit of g(x) as x approaches m from the left should equal the limit of g(x) as x approaches m from the right, g(m-) = g(m+).
• It must also be the case that g(m) equals these limits

In math terms, we need:

1. g(m-) = g(m+) = g(m)
   This translates to:

1. (m + 1) = m² = g(m)

Solving the equation

m + 1 = m²

gives us the continuous points of the function. Rearranging,

m² - m - 1 = 0

. This is a quadratic equation, and by solving we find that m could be any real number that satisfies this equation.