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QUESTION 5 Sketched below is the graph of f(x) = - x. 5.1 Write down the domain of f. Step 1: Identify the condition for the logarithm to be defined. For a logarithmic function x to be defined, its argument x must be strictly positive. The negative sign in front of x does not affect the domain. The domain of f is x > 0 5.2 Write down the equation of f^-1 in the form y = . Step 1: Assume a base for the logarithm. Given the context of part 5.3, where h(x) = 3^x - 5, it is implied that the base of the logarithm is 3. So, f(x) = -_3 x. Let y = -_3 x. Step 2: Swap x and y. x = -_3 y Step 3: Solve for y. Multiply both sides by -1: -x = _3 y Convert the logarithmic equation to an exponential equation: y = 3^-x The equation of f^-1(x) is y = 3^-x 5.3 Describe the transformation from f^-1 to h if h(x) = 3^x - 5. Step 1: Identify f^-1(x). From 5.2, f^-1(x) = 3^-x. Step 2: Compare f^-1(x) with h(x). We need to transform y = 3^-x into y = 3^x - 5. First, to change 3^-x to 3^x, we reflect the graph of y = 3^-x about the y-axis (replace x with -x). This gives y = 3^-(-x) = 3^x. Next, to change 3^x to 3^x - 5, we translate the graph of y = 3^x vertically downwards by 5 units. The transformation is a reflection about the y-axis followed by a vertical translation of 5 units downwards. 5.4 Use the graph of f to solve for x if k(x) is the reflection of f about the x-axis and k(x) ≥ 3. Step 1: Determine the equation of k(x). Given f(x) = -_3 x. A reflection about the x-axis