Given the function s(x)=4x,a) Write the equation for the function t(x) obtained by reflecting s(x) across the y-axis and shifting itdown by four units.b) Determine the domain and range of t(x).c) Sketch the graphs of s(x) and t(x) on the same set of axes.
Question
Given the function s(x)=4x,a) Write the equation for the function t(x) obtained by reflecting s(x) across the y-axis and shifting itdown by four units.b) Determine the domain and range of t(x).c) Sketch the graphs of s(x) and t(x) on the same set of axes.
Solution
a) Reflecting the function s(x) = 4x across the y-axis changes x to -x, so the function becomes s(-x) = 4(-x) = -4x. Shifting it down by four units means we subtract 4 from the function, so the final function t(x) is t(x) = -4x - 4.
b) The domain of t(x) is all real numbers, because you can substitute any real number for x and the function will still work. The range of t(x) is also all real numbers, because the function can output any real number depending on the input.
c) To sketch the graphs of s(x) and t(x), start by sketching s(x) = 4x. This is a straight line that passes through the origin (0,0) and has a slope of 4. Then, sketch t(x) = -4x - 4. This is also a straight line, but it has a slope of -4 and passes through the point (0,-4). The line for t(x) is a reflection of the line for s(x) across the y-axis, and is shifted down by 4 units.
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