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QUESTION 2 The graph of \( f \) is given by \( f(x)=3^{x} \) 2.1 Write down the asymptote of \( f \). 2.2 Determine the domain and range of \( f \). 2.3 Determine the equation of \( f^{-1} \) in the form \( y=\cdots \) \( 2.4 \quad \) Sketch the graph of \( f \) and \( f^{-1} \) on the same set of axes. 2.5 Write down the equation of \( g \) if \( g(x)=-f(x)+4 \) and describe the (2) transformation.

Ask by Schwartz Riley. in South Africa
Mar 17,2025

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**2.1 Asymptote** The graph of \( f(x)=3^{x} \) has a horizontal asymptote at \[ y=0 \] **2.2 Domain and Range** - **Domain:** All real numbers \[ \text{Domain}=\{x \in \mathbb{R}\} \] - **Range:** Positive real numbers \[ \text{Range}=(0,\infty) \] **2.3 Equation of \( f^{-1} \)** The inverse function is \[ f^{-1}(x)=\log_{3}x \] **2.4 Sketching the Graphs** - \( f(x)=3^{x} \) is an increasing curve passing through \((0,1)\) with a horizontal asymptote at \( y=0 \). - \( f^{-1}(x)=\log_{3}x \) is the reflection of \( f(x) \) over the line \( y=x \), passing through \((1,0)\) and \((3,1)\). **2.5 Equation of \( g \) and Transformation** \[ g(x)=-3^{x}+4 \] Transformation: 1. Reflect \( f(x) \) over the \( x \)-axis to get \(-3^{x}\). 2. Shift the graph upward by \( 4 \) units to obtain \( g(x)=-3^{x}+4 \).

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The function \( f(x) = 3^x \) has a horizontal asymptote at \( y = 0 \), which means that as \( x \) approaches negative infinity, the value of \( f(x) \) gets closer and closer to 0 but never actually reaches it. This characteristic is typical for exponential functions with a positive base: they rise steeply as \( x \) increases while leveling off near the x-axis. The domain of \( f \) is all real numbers \((-∞, +∞)\) since you can input any real number into the function and receive a corresponding output. The range, however, is \( (0, +∞) \) because exponential functions are always positive; that is, \( f(x) \) yields values greater than zero for all \( x \). For the inverse function \( f^{-1} \), we can express it as \( y = \log_3(x) \) after interchanging the variables and solving for \( y \).

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