Question

use transformations of the graph of f(x)= log x to graph the given function. graph and give the equation of the asymptote. use the graphs to determine the functions domain and range. g(x)=1 - log x determine the transformations that are needed to go from f(x)= log x to the given graph. choose the correct answer below. a. the graph of f(x)= log x should be reflected about the x-axis and vertically shifted 1 unit upward. b. the graph of f(x)= log x should be reflected about the x-axis and horizontally shifted 1 unit to the right. c. the graph of f(x)= log x should be reflected about the y-axis and horizontally shifted 1 unit to the left. d. the graph of f(x)= log x should be reflected about the y-axis and vertically shifted 1 unit downward.

Explanation:

Step1: Analyze the function transformation

Given \( f(x)=\log x \) and \( g(x) = 1-\log x\). We can rewrite \( g(x)\) as \( g(x)=-\log x + 1\).

• The negative sign in front of \( \log x \) indicates a reflection about the \( x \)-axis (since for a function \( y = f(x)\), \( y=-f(x)\) is a reflection over the \( x \)-axis).
• The \( + 1\) at the end indicates a vertical shift of 1 unit upward (since for a function \( y = f(x)\), \( y = f(x)+k\) is a vertical shift of \( k \) units, \( k>0\) is upward).

Now let's check the options:

• Option A: Reflect about \( x \)-axis (due to \(-\log x\)) and vertically shift 1 unit upward (due to \( + 1\)) - matches our analysis.
• Option B: Horizontal shift is not present here (the transformation is vertical shift, not horizontal), so B is wrong.
• Option C: Reflection about \( y \)-axis would be \( \log(-x)\), which is not the case here, and horizontal shift is incorrect, so C is wrong.
• Option D: Reflection about \( y \)-axis is incorrect, and vertical shift is upward not downward, so D is wrong.

Answer:

A. The graph of \( f(x)=\log x \) should be reflected about the x - axis and vertically shifted 1 unit upward.