Let \( f \) be the exponential function defined by \[ f(x)=10^{-(x-3)} \] and determine the following information based on \( f \). (i) Horizontal Asymptote: (ii) Range: (iii) \( y \)-intercept: (iv) Letter Corresponding to Graph:

Let's analyze the exponential function \( f(x) = 10^{-(x-3)} \) step by step to determine the required information. ### (i) Horizontal Asymptote For an exponential function of the form \( f(x) = a \cdot b^{x} + c \), the horizontal asymptote is determined by the constant \( c \). In the given function: \[ f(x) = 10^{-(x-3)} \] We can rewrite it as: \[ f(x) = 10^{-x + 3} = 10^{3} \cdot 10^{-x} = 1000 \cdot 10^{-x} \] As \( x \) approaches **infinity** (\( x \to \infty \)): \[ f(x) = 1000 \cdot 10^{-x} \to 0 \] As \( x \) approaches **negative infinity** (\( x \to -\infty \)): \[ f(x) = 1000 \cdot 10^{-x} \to \infty \] The function approaches **0** as \( x \) increases without bound, so the **horizontal asymptote** is: \[ y = 0 \] ### (ii) Range The exponential function \( 10^{-x} \) is always **positive** for all real numbers \( x \). Multiplying by 1000 (a positive constant) does not change this property. Therefore, the **range** of \( f(x) \) is: \[ (0, \infty) \] ### (iii) \( y \)-intercept The \( y \)-intercept occurs where \( x = 0 \). To find it, substitute \( x = 0 \) into the function: \[ f(0) = 10^{-(0-3)} = 10^{3} = 1000 \] So, the **\( y \)-intercept** is the point: \[ (0, 1000) \] ### (iv) Letter Corresponding to Graph *Without the accompanying graph options, it's not possible to determine which letter corresponds to the graph of the function.* --- **Summary:** 1. **Horizontal Asymptote:** \( y = 0 \) 2. **Range:** \( (0, \infty) \) 3. **\( y \)-intercept:** \( (0, 1000) \) 4. **Letter Corresponding to Graph:** *Cannot determine without graph options.*