This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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5.1 Write down the domain of f. The function is f(x) = -_3 x. For a logarithmic function _b x, the argument x must be greater than 0. The domain of f is x > 0. 5.2 Write down the equation of f^-1 in the form y=.... Step 1: Replace f(x) with y. y = -_3 x Step 2: Swap x and y to find the inverse. x = -_3 y Step 3: Isolate the logarithm. -x = _3 y Step 4: Convert the logarithmic equation to an exponential equation. y = 3^-x The equation of f^-1 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) = 3^x - 5. To change 3^-x to 3^x, we replace x with -x. This is a reflection in the y-axis. To change 3^x to 3^x - 5, we subtract 5 from the function. This is a vertical shift downwards by 5 units. The transformations are: • Reflection in the y-axis. • Vertical shift 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). The function f(x) = -_3 x. A reflection about the x-axis means multiplying the function by -1. k(x) = -f(x) = -(-_3 x) = _3 x Step 2: Set up the inequality k(x) 3. _3 x 3 Step 3: Solve the inequality for x. Convert the logarithmic inequality to an exponential inequality. Since the base 3 > 1, the inequality direction remains the same. x 3^3 x 27 Step 4: Consider the domain of k(x). For k(x) = _3 x, the domain requires x > 0. Combining x 27 with x > 0, the solution is x 27. The values of x are x 27. Last free one today — make it count tomorrow, or type /upgrade for unlimited.