Question

Domain and Range Suppose that \(a \neq 0, b \neq 1,\) and \(b>0\) . \((\mathbf{a}) y=a\left(b^{c-x}\right)+d \quad(\mathbf{b}) y=a \log _{b}(x-c)+d\)Determine the domain and range of the function.

Short answer

The domain and range of the first function, y=a(b^(c-x))+d, are \(D = R\) and either \(R_{>d}\) or \(R_{<d}\), depending on whether \(a\) is positive or negative. The domain and range of the second function, y=a*log_b(x-c)+d, are \(D = (c, +∞)\) and \(R = aR + d\), respectively.

Step-by-step solution

Find the domain of the first function

For the function y=a(b^(c-x))+d, since the exponent, c-x, can be any real number, the domain of the function is all real numbers, which can be written mathematically as \(D = R\)

Find the range of the first function

Since \(a \neq 0\) and \(b > 0\), it means that this function is always above or below \(d\), depending on whether \(a\) is positive or negative. If \(a > 0\), the function is always above \(d\); if \(a < 0\), the function is always below \(d\). Therefore, the range of the function is either \(R_{>d}\) or \(R_{<d}\), depending on whether \(a\) is positive or negative.

Find the domain of the second function

For the function y=a*log_b(x-c)+d, since logarithmic functions are defined only for positive numbers, \(x - c > 0\). Therefore, \(x > c\). So, the domain of the function is \(D = (c, +∞)\)

Find the range of the second function

The output of the log function can be any real number, and since \(a \neq 0\), the range of the function will be all real numbers, which can be written mathematically as \(R=aR+d\)