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Advanced Math Difficulty: Hard

The function $f$ is defined by $f (x) = a^{x} + b$ , where $a$ and $b$ are constants and $a greater than 0$ . In the xy-plane, the graph of $y = f (x)$ has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ and passes through the point $left parenthesis 2 comma 23 right parenthesis$ . What is the value of $a + b$ ?

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Explanation

The correct answer is $negative 19$ . It's given that function $f$ is defined by $f (x) = a^{x} + b$ , where $a$ and $b$ are constants and $a greater than 0$ . It's also given that the graph of $y = f (x)$ in the xy-plane has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ and passes through the point $left parenthesis 2 comma 23 right parenthesis$ . Since the graph has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ , $f (0) = - 25$ . Substituting $0$ for $x$ in the given equation yields $f (0) = a^{0} + b$ , or $f (0) = 1 + b$ , and substituting $negative 25$ for $f left parenthesis 0 right parenthesis$ in this equation yields $- 25 = 1 + b$ . Subtracting $1$ from each side of this equation yields $- 26 = b$ . Substituting $negative 26$ for $b$ in the equation $f (x) = a^{x} + b$ yields $f (x) = a^{x} - 26$ . Since the graph also passes through the point $left parenthesis 2 comma 23 right parenthesis$ , $f left parenthesis 2 right parenthesis equals 23$ . Substituting $2$ for $x$ in the equation $f (x) = a^{x} - 26$ yields $f (2) = a^{2} - 26$ , and substituting $23$ for $f left parenthesis 2 right parenthesis$ yields $23 = a^{2} - 26$ . Adding $26$ to each side of this equation yields $49 equals a squared$ . Taking the square root of both sides of this equation yields $\pm 7 = a$ . Since it's given that $a greater than 0$ , the value of $a$ is $7$ . It follows that the value of $a + b$ is $7 minus 26$ , or $negative 19$ .