What is the domain and range of f(x)=log_(4)(x+1)+2ir inequality notation? Domain Range all real numbers xgt -1 ygt -1 xleqslant -1 ygeqslant -1 xgt 2 ygt 2 xgeqslant 2 ygeqslant 2 x=4

To determine the domain and range of the function \( f(x) = \log_{4}(x+1) + 2 \), we need to analyze the behavior of the logarithmic function.<br /><br />### Domain:<br />The argument of the logarithm, \( x + 1 \), must be greater than zero because the logarithm is only defined for positive values. Therefore, we have:<br />\[ x + 1 > 0 \]<br />\[ x > -1 \]<br /><br />So, the domain of \( f(x) \) is:<br />\[ x > -1 \]<br /><br />### Range:<br />The range of a logarithmic function \( \log_{4}(x+1) \) is all real numbers because as \( x \) approaches \(-1\) from the right, \( \log_{4}(x+1) \) approaches \(-\infty\), and as \( x \) increases without bound, \( \log_{4}(x+1) \) increases without bound.<br /><br />Adding 2 to the logarithmic function shifts the entire range up by 2 units. Therefore, the range of \( f(x) = \log_{4}(x+1) + 2 \) is:<br />\[ y > 2 \]<br /><br />So, the correct answers are:<br />- Domain: \( x > -1 \)<br />- Range: \( y > 2 \)