(i) Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base and 'x' is the variable. Logarithmic functions are the inverse of exponential functions and are of the form f(x) = log_b(x), where 'b' is the base and 'x' is the variable. The key factors in these functions are the base 'b' and the variable 'x'. The domain of an exponential function is all real numbers, and the range is all positive real numbers. The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

(ii) Exponential functions grow by equal factors over equal intervals. Logarithmic functions grow by unequal factors over equal intervals. Power functions grow at a constant rate. For example, f(x) = 2^x is an exponential function, f(x) = log_2(x) is a logarithmic function, and f(x) = x^2 is a power function. The exponential function has a horizontal asymptote at y=0, the logarithmic function has a vertical asymptote at x=0, and the power function has no asymptotes. The exponential and power functions have an x-intercept at x=0, while the logarithmic function does not have an x-intercept.

(iii) A function has exponential growth if it increases by a constant factor over equal intervals. This can be seen in the graph of the function as a curve that becomes steeper and steeper as x increases.

(iv) Between exponential and logarithmic functions, the exponential function grows faster. This is because the exponential function increases by a constant factor over equal intervals, while the logarithmic function increases by unequal factors over equal intervals.

(v) The growth patterns and special points of the functions can be observed in their graphs. The exponential function has a horizontal asymptote at y=0 and an x-intercept at x=0. The logarithmic function has a vertical asymptote at x=0 and no x-intercept. The power function has no asymptotes and an x-intercept at x=0. The exponential function becomes steeper and steeper as x increases, indicating exponential growth. The logarithmic function becomes less steep as x increases, indicating slower growth. The power function grows at a constant rate.

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(i) Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base and 'x' is the variable. Logarithmic functions are the inverse of exponential functions and are of the form f(x) = log_b(x), where 'b' is the base and 'x' is the variable. The key factors in these functions are the base 'b' and the variable 'x'. The domain of an exponential function is all real numbers, and the range is all positive real numbers. The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

(ii) Exponential functions grow by equal factors over equal intervals. Logarithmic functions grow by unequal factors over equal intervals. Power functions grow at a constant rate. For example, f(x) = 2^x is an exponential function, f(x) = log_2(x) is a logarithmic function, and f(x) = x^2 is a power function. The exponential function has a horizontal asymptote at y=0, the logarithmic function has a vertical asymptote at x=0, and the power function has no asymptotes. The exponential and power functions have an x-intercept at x=0, while the logarithmic function does not have an x-intercept.

(iii) A function has exponential growth if it increases by a constant factor over equal intervals. This can be seen in the graph of the function as a curve that becomes steeper and steeper as x increases.

(iv) Between exponential and logarithmic functions, the exponential function grows faster. This is because the exponential function increases by a constant factor over equal intervals, while the logarithmic function increases by unequal factors over equal intervals.

(v) The growth patterns and special points of the functions can be observed in their graphs. The exponential function has a horizontal asymptote at y=0 and an x-intercept at x=0. The logarithmic function has a vertical asymptote at x=0 and no x-intercept. The power function has no asymptotes and an x-intercept at x=0. The exponential function becomes steeper and steeper as x increases, indicating exponential growth. The logarithmic function becomes less steep as x increases, indicating slower growth. The power function grows at a constant rate.

Knowee AI · Accepted Answer

(i) Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base and 'x' is the variable. Logarithmic functions are the inverse of exponential functions and are of the form f(x) = log_b(x), where 'b' is the base and 'x' is the variable. The key factors in these functions are the base 'b' and the variable 'x'. The domain of an exponential function is all real numbers, and the range is all positive real numbers. The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

(ii) Exponential functions grow by equal factors over equal intervals. Logarithmic functions grow by unequal factors over equal intervals. Power functions grow at a constant rate. For example, f(x) = 2^x is an exponential function, f(x) = log_2(x) is a logarithmic function, and f(x) = x^2 is a power function. The exponential function has a horizontal asymptote at y=0, the logarithmic function has a vertical asymptote at x=0, and the power function has no asymptotes. The exponential and power functions have an x-intercept at x=0, while the logarithmic function does not have an x-intercept.

(iii) A function has exponential growth if it increases by a constant factor over equal intervals. This can be seen in the graph of the function as a curve that becomes steeper and steeper as x increases.

(iv) Between exponential and logarithmic functions, the exponential function grows faster. This is because the exponential function increases by a constant factor over equal intervals, while the logarithmic function increases by unequal factors over equal intervals.

(v) The growth patterns and special points of the functions can be observed in their graphs. The exponential function has a horizontal asymptote at y=0 and an x-intercept at x=0. The logarithmic function has a vertical asymptote at x=0 and no x-intercept. The power function has no asymptotes and an x-intercept at x=0. The exponential function becomes steeper and steeper as x increases, indicating exponential growth. The logarithmic function becomes less steep as x increases, indicating slower growth. The power function grows at a constant rate.

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