91Ó°ÊÓ

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{3}-x+3$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|-2 x+5|=|x+3|$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(2 x)$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|-5 x+1|=|3 x-4|$$

Concept Check Suppose that the graph of \(y=x^{2}\) is translated in such a way that its domain is \((-\infty, \infty)\) and its range is \([38, \infty) .\) What values of \(h\) and \(k\) can be used if the new function is of the form \(y=(x-h)^{2}+k ?\)

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=-3 x, g(x)=-\frac{1}{3} x$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$2 f(x-1)$$

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{4}-5 x+2$$

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$3 f\left(\frac{1}{4} x\right)$$