1. **State the problem:** We are given two functions: \(f(x) = \left(\frac{3}{4}\right)^x\) and \(g(x)\) represented by a graph. We need to determine which statement about \(f(x)\) and \(g(x)\) is true from the options A, B, C, and D. 2. **Analyze \(f(x)\):** The function \(f(x) = \left(\frac{3}{4}\right)^x\) is an exponential function with base \(\frac{3}{4} < 1\). - Since the base is between 0 and 1, \(f(x)\) is a decreasing function on its entire domain. - The horizontal asymptote of \(f(x)\) is \(y = 0\) because as \(x \to \infty\), \(f(x) \to 0\). - The y-intercept is found by evaluating \(f(0) = \left(\frac{3}{4}\right)^0 = 1\). 3. **Analyze \(g(x)\) from the graph description:** - The graph passes through approximately \((0,1)\), so the y-intercept is \(1\). - It rises to a local maximum near \((1,2)\), then falls steeply crossing the x-axis near \(x=2\) and continues downward. - Since it crosses the x-axis and goes downward without bound, \(g(x)\) is not decreasing on its entire domain. - The graph does not approach \(-\infty\) as \(x \to \infty\) because it falls steeply downward, so it does approach \(-\infty\). - The graph does not have a horizontal asymptote of \(y=0\) because it crosses the x-axis and continues downward. 4. **Evaluate each statement:** - A. Both functions are decreasing on their entire domains. - \(f(x)\) is decreasing, but \(g(x)\) is not (it has a local maximum and then decreases). - So, A is false. - B. Both functions approach \(-\infty\) as \(x\) approaches infinity. - \(f(x)\) approaches 0, not \(-\infty\). - \(g(x)\) approaches \(-\infty\). - So, B is false. - C. Both functions have an asymptote of \(y=0\). - \(f(x)\) has a horizontal asymptote at \(y=0\). - \(g(x)\) does not have a horizontal asymptote at \(y=0\) because it crosses the x-axis and continues downward. - So, C is false. - D. Both functions have a y-intercept of \((0,1)\). - \(f(0) = 1\). - \(g(x)\) passes through \((0,1)\). - So, D is true. **Final answer:** \(\boxed{\text{D. Both functions have a y-intercept of } (0,1)}\)