7/13 Which of the following statements are true about the function below? Select all that apply. \[ f(x)=0.5(2)^{x}-1 \] Select all correct options The \( y \)-intercept is \( (0,0.5) \) The function represents exponential growth The horizontal asymptote is \( \mathrm{y}=-1 \) The domain is all real numbers \( (-\infty, \infty) \) The range is all real numbers \( (-\infty, \infty) \)

To analyze the function \( f(x) = 0.5(2)^{x} - 1 \), we will evaluate each statement one by one. 1. **The \( y \)-intercept is \( (0,0.5) \)**: - To find the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = 0.5(2)^{0} - 1 = 0.5(1) - 1 = 0.5 - 1 = -0.5 \] - Therefore, the \( y \)-intercept is \( (0, -0.5) \), not \( (0, 0.5) \). This statement is **false**. 2. **The function represents exponential growth**: - The function has the form \( a(b)^{x} + c \) where \( a = 0.5 \), \( b = 2 \) (which is greater than 1), and \( c = -1 \). Since \( b > 1 \), the function indeed represents exponential growth. This statement is **true**. 3. **The horizontal asymptote is \( y = -1 \)**: - As \( x \) approaches infinity, \( (2)^{x} \) grows without bound, and thus \( f(x) \) approaches \( \infty \). As \( x \) approaches negative infinity, \( (2)^{x} \) approaches \( 0 \): \[ \lim_{x \to -\infty} f(x) = 0.5(0) - 1 = -1 \] - Therefore, the horizontal asymptote is indeed \( y = -1 \). This statement is **true**. 4. **The domain is all real numbers \( (-\infty, \infty) \)**: - The function \( f(x) = 0.5(2)^{x} - 1 \) is defined for all real numbers \( x \). Thus, the domain is indeed all real numbers. This statement is **true**. 5. **The range is all real numbers \( (-\infty, \infty) \)**: - The function approaches \( -1 \) as \( x \) approaches negative infinity and grows without bound as \( x \) approaches positive infinity. Therefore, the range is \( (-1, \infty) \), not all real numbers. This statement is **false**. ### Summary of Statements: - The \( y \)-intercept is \( (0,0.5) \): **False** - The function represents exponential growth: **True** - The horizontal asymptote is \( y = -1 \): **True** - The domain is all real numbers \( (-\infty, \infty) \): **True** - The range is all real numbers \( (-\infty, \infty) \): **False** ### Correct Options: - The function represents exponential growth - The horizontal asymptote is \( y = -1 \) - The domain is all real numbers \( (-\infty, \infty) \)