Identify the y-intercept.

5.1 Write down the coordinates of B. Step 1: Identify the y-intercept. B is the y-intercept of the function g(x) = a^x. To find the y-intercept, set x=0. g(0) = a^0 = 1 The coordinates of B are (0, 1) 5.2 Determine the value of a. Step 1: Substitute the coordinates of point A into the function. Point A (-3, (27)/(8)) lies on g(x) = a^x. (27)/(8) = a^-3 Step 2: Solve for a. (27)/(8) = (1)/(a^3) a^3 = (8)/(27) Take the cube root of both sides: a = 3/(27) a = (2)/(3) The value of a is (2)/(3) 5.3 Determine the equation of h if h is the reflection of g in the y-axis. Step 1: Apply the reflection rule. A reflection in the y-axis means replacing x with -x in the function. Given g(x) = ((2)/(3))^x. h(x) = g(-x) h(x) = ((2)/(3))^-x Step 2: Simplify the expression. h(x) = ((3)/(2))^x The equation of h is h(x) = ((3)/(2))^x 5.4 Write down the equation of h^-1(x), the inverse of h, in the form y = . Step 1: Start with the equation for h(x). y = ((3)/(2))^x Step 2: Swap x and y to find the inverse. x = ((3)/(2))^y Step 3: Solve for y by converting to logarithmic form. y = _(3)/(2) x The equation of h^-1(x) is y = _(3)/(2) x 5.5 For which values of x will h^-1(x) ≤ 1? Step 1: Set up the inequality using the inverse function. _(3)/(2) x ≤ 1 Step 2: Convert the logarithmic inequality to an exponential inequality. Since the base b = (3)/(2) is greater than 1, the inequality direction remains the same. x ≤ ((3)/(2))^1 x ≤ (3)/(2) Step 3: Consider the domain of the logarithmic function. The argument of a logarithm must be positive. x > 0 Combining the conditions, the values of x are 0 < x ≤ (3)/(2) 5.6 Write down the domain of h^-1(x - 3). Step 1: Recall the domain of h^-1(x). The domain of h^-1(x) = _(3)/(2) x is x > 0. Step 2: Apply the domain condition to h^-1(x - 3). For h^-1(x - 3), the argument of the logarithm is (x - 3). This argument must be greater than 0. x - 3 > 0 x > 3 The domain of h^-1(x - 3) is x > 3 That's 2 down. 3 left today — send the next one.